Particle Physics Reference Library: Volume 2: Detectors for Particles and Radiation [1st ed.] 9783030353179, 9783030353186

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Christian W. Fabjan Herwig Schopper Editors

Particle Physics Reference Library Volume 2: Detectors for Particles and Radiation

Particle Physics Reference Library

Christian W. Fabjan • Herwig Schopper Editors

Particle Physics Reference Library Volume 2: Detectors for Particles and Radiation

Editors Christian W. Fabjan Austrian Academy of Sciences and University of Technology Vienna, Austria

ISBN 978-3-030-35317-9 https://doi.org/10.1007/978-3-030-35318-6

Herwig Schopper CERN Geneva, Switzerland

ISBN 978-3-030-35318-6 (eBook)

This book is an open access publication. © The Editor(s) (if applicable) and The Author(s) 2011, 2020 Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

For many years, the Landolt-Börnstein—Group I Elementary Particles, Nuclei and Atoms, Vol. 21A (Physics and Methods. Theory and Experiments, 2008), Vol. 21B1 (Elementary Particles. Detectors for Particles and Radiation. Part 1: Principles and Methods, 2011), Vol. 21B2 (Elementary Particles. Detectors for Particles and Radiation. Part 2: Systems and Applications), and Vol. 21C (Elementary Particles. Accelerators and Colliders, 2013), has served as a major reference work in the field of high-energy physics. When, not long after the publication of the last volume, open access (OA) became a reality for HEP journals in 2014, discussions between Springer and CERN intensified to find a solution for the “Labö” which would make the content available in the same spirit to readers worldwide. This was helped by the fact that many researchers in the field expressed similar views and their readiness to contribute. Eventually, in 2016, on the initiative of Springer, CERN and the original Labö volume editors agreed in tackling the issue by proposing to the contributing authors a new OA edition of their work. From these discussions a compromise emerged along the following lines: transfer as much as possible of the original material into open access; add some new material reflecting new developments and important discoveries, such as the Higgs boson; and adapt to the conditions due to the change from copyright to a CC BY 4.0 license. Some authors were no longer available for making such changes, having either retired or, in some cases, deceased. In most such cases, it was possible to find colleagues willing to take care of the necessary revisions. A few manuscripts could not be updated and are therefore not included in this edition. We consider that this new edition essentially fulfills the main goal that motivated us in the first place—there are some gaps compared to the original edition, as explained, as there are some entirely new contributions. Many contributions have been only minimally revised in order to make the original status of the field available as historical testimony. Others are in the form of the original contribution being supplemented with a detailed appendix relating recent developments in the field. However, a substantial fraction of contributions has been thoroughly revisited by their authors resulting in true new editions of their original material. v

vi

Preface

We would like to express our appreciation and gratitude to the contributing authors, to the colleagues at CERN involved in the project, and to the publisher, who has helped making this very special endeavor possible. Vienna, Austria Geneva, Switzerland Geneva, Switzerland July 2020

Christian W. Fabjan Stephen Myers Herwig Schopper

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Christian W. Fabjan and Herwig Schopper

1

2

The Interaction of Radiation with Matter . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Hans Bichsel and Heinrich Schindler

5

3

Scintillation Detectors for Charged Particles and Photons .. . . . . . . . . P. Lecoq

45

4

Gaseous Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . H. J. Hilke and W. Riegler

91

5

Solid State Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . G. Lutz and R. Klanner

137

6

Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . C. W. Fabjan and D. Fournier

201

7

Particle Identification: Time-of-Flight, Cherenkov and Transition Radiation Detectors . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Roger Forty and Olav Ullaland

281

8

Neutrino Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Leslie Camilleri

337

9

Nuclear Emulsions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Akitaka Ariga, Tomoko Ariga, Giovanni De Lellis, Antonio Ereditato, and Kimio Niwa

383

10 Signal Processing for Particle Detectors . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . V. Radeka

439

11 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . J. Apostolakis

485

vii

viii

Contents

12 Triggering and High-Level Data Selection .. . . . . . . . . .. . . . . . . . . . . . . . . . . . W. H. Smith

533

13 Pattern Recognition and Reconstruction . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . R. Frühwirth, E. Brondolin, and A. Strandlie

555

14 Distributed Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Manuel Delfino

613

15 Statistical Issues in Particle Physics . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Louis Lyons

645

16 Integration of Detectors into a Large Experiment: Examples from ATLAS and CMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Daniel Froidevaux

693

17 Neutrino Detectors Under Water and Ice . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Christian Spiering

785

18 Spaceborne Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Roberto Battiston

823

19 Cryogenic Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Klaus Pretzl

871

20 Detectors in Medicine and Biology . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . P. Lecoq

913

21 Solid State Detectors for High Radiation Environments . . . . . . . . . . . . . Gregor Kramberger

965

22 Future Developments of Detectors .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . Ties Behnke, Karsten Buesser, and Andreas Mussgiller

1035

About the Editors

Christian W. Fabjan is an experimental particle physicist, who spent the major part of his career at CERN, with leading involvement in several of the major CERN programs. At the Intersecting Storage Rings, he concentrated on strong interaction physics and the development of new experimental techniques and followed at the Super Synchrotron with experiments in the Relativistic Heavy Ion program. At the Large Hadron Collider, he focused on the development of several experimental techniques and participated in the ALICE experiment as Technical Coordinator. He is affiliated with the Vienna University of Technology and was, most recently, leading the institute of High Energy Physics of the Austrian Academy of Sciences. Herwig Schopper joined as a research associate at CERN since 1966 and returned in 1970 as leader of the Nuclear Physics Division and went on to become a member of the directorate responsible for the coordination of CERN’s experimental program. He was chairman of the ISR Committee at CERN from 1973 to 1976 and was elected as member of the Scientific Policy Committee in 1979. Following Léon Van Hove and John Adams’ years as Director-General for research and executive Director-General, Schopper became the sole Director-General of CERN in 1981. Schopper’s years as CERN’s Director-General saw the construction and installation of the Large ElectronPositron Collider (LEP) and the first tests of four detectors for the LEP experiments. Several facilities (including ISR, BEBC, and EHS) had to be closed to free up resources for LEP. ix

Chapter 1

Introduction Christian W. Fabjan and Herwig Schopper

Enormous progress has been achieved during the last three decades in the understanding of the microcosm. This was possible by a close interplay between new theoretical ideas and precise experimental data. The present state of our knowledge has been summarised in Volume I/21A “Theory and Experiments”. This Volume I/21B is devoted to detection methods and techniques and data acquisition and handling. The rapid increase of our knowledge of the microcosm was possible only because of an astonishingly fast evolution of detectors for particles and photons. Since the early days of scintillation screens and Geiger counters a series of completely new detector concepts was developed. They are based on imaginative ideas, sometimes even earning a Nobel Prize, combined with sophisticated technological developments. It might seem surprising that the exploration of an utterly abstract domain like particle physics, requires the most advanced techniques, but this makes the whole field so attractive. The development of detectors was above all pushed by the requirements of particle physics. In order to explore smaller structures one has to use finer probes, i.e. shorter wavelengths implying higher particle energies. This requires detectors for high-energy particles and photons. At the same time one has to cope with the quantum-mechanical principle that cross sections for particle interactions have a tendency to fall with increasing interaction energy. Therefore accelerators or colliders have to deliver not only higher energies but at the same time also higher collision rates. This implies that detectors must sustain higher rates. This problem is aggravated by the fact that the high-energy frontier is at present linked to hadron

C. W. Fabjan () Austrian Academy of Sciences and University of Technology, Vienna, Austria e-mail: [email protected] H. Schopper CERN, Geneva, Switzerland © The Author(s) 2020 C. W. Fabjan, H. Schopper (eds.), Particle Physics Reference Library, https://doi.org/10.1007/978-3-030-35318-6_1

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C. W. Fabjan and H Schopper.

collisions. Electron-positron colliders are characterised by events with relatively few outgoing particles since two pointlike particles collide and the strong interaction is negligible in such reactions. After the shutdown of LEP in 2000 the next electronpositron collider is far in the future and progress is now depending on proton-proton collisions at the LHC at CERN or heavy ion colliders, e.g. GSI, Germany, RHIC at BNL in the USA and also LHC. Protons are composite particles containing quarks and gluons and hence proton collisions produce very complicated events with many hundreds of particles. Consequently, detectors had to be developed which are able to cope with extremely high data rates and have to resist high levels of irradiation. Such developments were in particular motivated by the needs of the LHC experiments. It seems plausible that accelerators and colliders have to grow in size with increasing energy. But why have detectors to be so large? Their task is to determine the direction of emitted particles, measure their momenta or energy and in some cases their velocity which together with the momentum allows to determine their mass and hence to identify the nature of the particle. The most precise method to measure the momentum of charged particles is to determine their deflection in a magnetic field which is proportional to B · l where B is the magnetic field strength and l the length of the trajectory in the magnetic field. Of course, it is also determined by the spatial resolution of the detector to determine the track. To attain the highest possible precision superconducting coils are used in most experiments to produce a large B. Great efforts have been made to construct detectors with a spatial resolution down to the order of several microns. But even then track lengths l of the order of several meters are needed to measure momenta with a precision of about 1% of particles with momenta of several 100 GeV/c. This is the main reason why experiments must have extensions of several meters and weigh thousands of tons. Another possibility to determine the energy of particles are so-called “calorimeters”. This name is misleading since calorimeters have nothing to do with calorific measurements but this name became ubiquitous to indicate that the total energy of a particle is measured. The measurement is done in the following way. A particle hits the material of the detector, interacts with an atom, produces secondary particles which, if sufficiently energetic, generate further particles, leading to a whole cascade of particles of ever decreasing energies. The energy deposited in the detector material can be measured in various ways. If the material of the detector is a scintillator (crystal, liquid or gas), the scintillating light is approximately proportional to the deposited energy and it can be observed by, e.g., photomultipliers. Alternatively, the ionisation produced by the particle cascade can be measured by electrical means. In principle two kinds of calorimeters can be distinguished. Electrons and photons produce a so-called electromagnetic cascade due to electromagnetic interactions. Such cascades are relatively small both in length and in lateral dimension. Hence electromagnetic calorimeters can consist of a homogenous detector material containing the whole cascade. Incident hadrons, however, produce in the cascade also a large number of neutrons which can travel relatively long ways before losing their energy and therefore hadronic cascades have large geometrical extensions even

1 Introduction

3

in the densest materials (of the order few meters in iron). Therefore the detectors for hadronic cascades are composed of a sandwich of absorber material interspersed with elements to detect the deposited energy. In such a device, only a certain fraction of the total energy is sampled. The challenge of the design consists in making this fraction as much as possible proportional to the total energy. The main advantage of calorimeters, apart from the sensitivity to both charged and neutral particles, is that their size increases only logarithmically with the energy of the incident particle, hence much less than for magnetic spectrometers, albeit with an energy resolution inferior to magnetic spectrometers below about 100 GeV. They require therefore comparatively little space which is of paramount importance for colliders where the solid angle around the interaction area has to be covered in most cases as fully as possible. Other detectors have been developed for particular applications, e.g. for muon and neutrino detection or the observation of cosmic rays in the atmosphere or deep underground/water. Experiments in space pose completely new problems related to mechanical stability and restrictions on power consumption and consumables. The main aim in the development of all these detectors is higher sensitivity, better precision and less influence by the environment. Obviously, reduction of cost has become a major issue in view of the millions of detector channels in most modern experiments. New and more sophisticated detectors need better signal processing, data acquisition and networking. Experiments at large accelerators and colliders pose special problems dictated by the beam properties and restricted space. Imagination is the key to overcome such challenges. Experiments at accelerators/colliders and for the observation of cosmic rays have become big projects involving hundreds or even thousands of scientists and the time from the initial proposal to data taking may cover one to two decades. Hence it is sometimes argued that they are not well adapted for the training of students. However, the development of a new detector is subdivided in a large number of smaller tasks (concept of the detector, building prototypes, testing, computer simulations and preparation of the data acquisition), each lasting only a few years and therefore rather well suited for a master or PhD thesis. The final “mass production” of many detection channels in the full detector assembly, however, is eventually transferred to industry. These kinds of activities may in some cases have little to do with particle physics itself, but they provide an excellent basis for later employment in industry. Apart from specific knowledge, e.g., in vacuum, magnets, gas discharges, electronics, computing and networking, students learn how to work in the environment of a large project respecting time schedules and budgetary restrictions—and perhaps even most important to be trained to work in an international environment. Because the development of detectors does not require the resources of a large project but can be carried out in a small laboratory, most of these developments are done at universities. Indeed most of the progress in detector development is due to universities or national laboratories. However, when it comes to plan a large experiment these originally individual activities are combined and coordinated

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C. W. Fabjan and H Schopper.

which naturally leads to international cooperation between scientists from different countries, political traditions, creeds and mentalities. To learn how to adapt to such an international environment represents a human value which goes much beyond the scientific achievements. The stunning success of the “Standard Model of particle physics” also exhibits with remarkable clarity its limitations. The many open fundamental issues— origin of CP-violation, neutrino mass, dark matter and dark energy, to name just few—are motivating a vast, multi-faceted research programme for accelerator- and non-accelerator based, earth- and space-based experimentation. This has led to a vigorous R&D in detectors and data handling. This revised edition provides an update on these developments over the past 7–9 years. We gratefully acknowledge the very constructive collaboration with the authors of the first edition, in several cases assisted by additional authors. May this Open Access publication reach a global readership, for the benefit of science.

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made. The images or other third party material in this chapter are included in the chapter’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Chapter 2

The Interaction of Radiation with Matter Hans Bichsel and Heinrich Schindler

2.1 Introduction The detection of charged particles is usually based on their electromagnetic interactions with the electrons and nuclei of a detector medium. Interaction with the Coulomb field of the nucleus leads to deflections of the particle trajectory (multiple scattering) and to radiative energy loss (bremsstrahlung). Since the latter, discussed in Sect. 2.4.1, is inversely proportional to the particle mass squared, it is most significant for electrons and positrons. “Heavy” charged particles (in this context: particles with a mass M exceeding the electron mass m) passing through matter lose energy predominantly through collisions with electrons. Our theoretical understanding of this process, which has been summarised in a number of review articles [1–7] and textbooks [8–13], is based on the works of some of the most prominent physicists of the twentieth century, including Bohr [14, 15], Bethe [16, 17], Fermi [18, 19], and Landau [20]. After outlining the quantum-mechanical description of single collisions in terms of the double-differential cross section d2 σ/ (dEdq), where E and q are the energy transfer and momentum transfer involved in the collision, Sect. 2.3 discusses algorithms for the quantitative evaluation of the single-differential cross section

The author Hans Bichsel is deceased at the time of publication. H. Bichsel · H. Schindler () CERN, Geneva, Switzerland e-mail: [email protected] © The Author(s) 2020 C. W. Fabjan, H. Schopper (eds.), Particle Physics Reference Library, https://doi.org/10.1007/978-3-030-35318-6_2

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H. Bichsel and H. Schindler

dσ/dE and its moments. The integral cross section (zeroth moment), multiplied by the atomic density N, corresponds to the charged particle’s inverse mean free path λ−1 or, in other words, the average number of collisions per unit track length, −1

λ

E max

= M0 = N

dσ dE. dE

(2.1)

Emin

The stopping power dE/dx, i.e. the average energy loss per unit track length, is given by the first moment, dE − = M1 = N dx

E max

E

dσ dE. dE

(2.2)

Emin

The integration limits Emin, max are determined by kinematics. Due to the stochastic nature of the interaction process, the number of collisions and the sum  of energy losses along a particle track are subject to fluctuations. Section 2.5 deals with methods for calculating the probability density distribution f (, x) for different track lengths x. The energy transfer from the incident particle to the electrons of the medium typically results in excitation and ionisation of the target atoms. These observable effects are discussed in Sect. 2.6. As a prologue to the discussion of charged-particle collisions, Sect. 2.2 briefly reviews the principal photon interaction mechanisms in the X-ray and gamma ray energy range. Throughout this chapter, we attempt to write all expressions in a way independent of the system of units (cgs or SI), by using the fine structure constant α ∼ 1/137. Other physical constants used occasionally in this chapter include the  Rydberg  energy Ry = α 2 mc2 /2 ∼ 13.6 eV, and the Bohr radius a0 = h¯ c/ αmc2 ∼ 0.529 Å. Cross-sections are quoted in barn (1 b = 10−24 cm2 ).

2.2 Photon Interactions Photons interact with matter via a range of mechanisms, which can be classified according to the type of target, and the effect of the interaction on the photon (absorption or scattering) [9, 21]. At energies beyond the ultraviolet range, the dominant processes are photoelectric absorption (Sect. 2.2.1), Compton scattering (Sect. 2.2.2), and pair production (Sect. 2.2.3). As illustrated in Fig. 2.1, photoabsorption constitutes the largest contribution to the total cross section at low photon energies, pair production is the most frequent interaction at high energies, and Compton scattering dominates in the intermediate energy range.

2 The Interaction of Radiation with Matter

102

E [MeV]

Fig. 2.1 The lower curve shows, as a function of the atomic number Z of the target material, the photon energy E below which photoelectric absorption is the most probable interaction mechanism, while the upper curve shows the energy above which pair production is the most important process. The shaded region between the two curves corresponds to the domain where Compton scattering dominates. The cross sections are taken from the NIST XCOM database [24]

7

pair production 10

Compton scattering

1

10–1 photoabsorption –2

10

10–3

20

40

60

80 Z

Detailed descriptions of these processes can be found, for instance, in Refs. [8– 10, 12, 22, 23]. The focus of this section is on photoabsorption, the description of which (as will be discussed in Sect. 2.3) is related to that of inelastic charged particle collisions in the regime of low momentum transfer.

2.2.1 Photoabsorption In a photoelectric absorption interaction, the incident photon disappears and its energy is transferred to the target atom (or group of atoms). The intensity I of a monochromatic beam of photons with energy E thus decreases exponentially as a function of the penetration depth x in a material, I (x) = I0 e−μx , where the attenuation coefficient μ is proportional to the atomic density N of the medium and the photoabsorption cross section σγ , μ (E) = Nσγ (E) . Let us first consider a (dipole-allowed) transition between the ground state |0 of an atom and a discrete excited state |n with excitation energy En . The integral photoabsorption cross section of the line is given by  2π 2 α (h¯ c)2 σγ(n) (E) dE = fn . mc2

8

H. Bichsel and H. Schindler

The dimensionless quantity fn =

Z  E |n| ri |0|2 , 2 n 3 (hc) ¯ j =1

2mc2

(2.3)

with the sum extending over the electrons in the target atom, is known as the dipole oscillator strength (DOS). Similarly, transitions to the continuum are characterised by the dipole oscillator strength density df/dE, and the photoionisation cross section σγ (E) is given by σγ (E) =

2π 2 α (h¯ c)2 df (E) . dE mc2

(2.4)

The dipole oscillator strength satisfies the Thomas-Reiche-Kuhn (TRK) sum rule, 

 fn +

n

dE

df (E) = Z. dE

(2.5)

σγ [Mb]

 For most gases, the contribution of excited states ( fn ) to the TRK sum rule is a few percent of the total, e.g. ∼5% for argon and ∼7% for methane [25, 26]. As can be seen from Fig. 2.2, the photoabsorption cross section reflects the atomic shell structure. Evaluated atomic and molecular photoabsorption cross

102 10

Ne

1

Ar

10–1 10–2 10–3 10–4

102

103

104 E [eV]

Fig. 2.2 Photoabsorption cross sections of argon (solid curve) and neon (dashed curve) as a function of the photon energy E [25, 26]

2 The Interaction of Radiation with Matter

9

sections (both for discrete excitations as well as transitions to the continuum) for many commonly used gases are given in the book by Berkowitz [25, 26]. At energies sufficiently above the ionisation threshold, the molecular photoabsorption cross section is, to a good approximation, given by the sum of the photoabsorption cross sections of the constituent atoms. A comprehensive compilation of atomic photoabsorption data (in the energy range between ∼30 eV and 30 keV) can be found in Ref. [27]. Calculations for energies between 1 and 100 GeV are available in the NIST XCOM database [24]. Calculated photoionisation cross sections for individual shells can be found in Refs. [28–30]. At high energies, i.e. above the respective absorption edges, photons interact preferentially with innershell electrons. The subsequent relaxation processes (emission of fluorescence photons and Auger electrons) are discussed in Sect. 2.6. The response of a solid with atomic number Z to an incident photon of energy E = h¯ ω is customarily described in terms of the complex dielectric function ε(ω) = ε1 (ω) + iε2 (ω). The oscillator strength density is related to ε(ω) by   2Z ε2 (E) −1 df (E) 2Z =E  , Im = E 2 2 2  2 dE ε (E) π h¯ p ε1 (E) + ε2 (E) π h

¯ p

(2.6)

where h¯ p =

4πα (h¯ c)3 NZ mc2

(2.7)

is the plasma energy of the material, which depends only on the electron density NZ. In terms of the dielectric loss function Im (−1/ε), the TRK sum rule reads 



 2 −1 π dE Im E= h¯ p . ε (E) 2

(2.8)

Compilations of evaluated optical data for semiconductors are available in Ref. [32], and for solids in general in Ref. [31]. As an example, Fig. 2.3 shows the dielectric loss function of silicon, a prominent feature of which is the peak at ∼17 eV, corresponding to the plasma energy of the four valence (M-shell) electrons.

2.2.2 Compton Scattering Compton scattering refers to the collision of a photon with a weakly bound electron, whereby the photon transfers part of its energy to the electron and is deflected with respect to its original direction of propagation. We assume in the following that the target electron is free and initially at rest, which is a good approximation if the photon energy E is large compared to the electron’s binding energy. Due to

10

H. Bichsel and H. Schindler

Im(-1/ε)

Fig. 2.3 Dielectric loss function Im (−1/ε (E)) of solid silicon [31] as a function of the photon energy E

10

plasmon peak

1

L 23 edge

10−1 10−2 10−3 10−4

K edge

band gap

10−5 10−6 10−7 10−1

1

10

102

103 104 E [eV]

conservation of energy and momentum, the photon energy E  after the collision and the scattering angle θ of the photon are then related by E =

mc2 , 1 − cos θ + (1/u)

(2.9)

  where u = E/ mc2 is the photon energy (before the collision) in units of the electron rest energy. The kinetic energy T = E − E  imparted to the electron is largest for a headon collision (θ = π) and the energy spectrum of the recoil electrons consequently exhibits a cut-off (Compton edge) at Tmax = E

2u . 1 + 2u

The total cross section (per electron) for the Compton scattering of an unpolarised photon by a free electron at rest, derived by Klein and Nishina in 1929 [33], is given by  σ (KN) = 2π

 2 

 α hc 1 + u 2 (1 + u) ln (1 + 2u) ln (1 + 2u) 1 + 3u ¯ − + − . 1 + 2u u 2u mc2 u2 (1 + 2u)2

(2.10) At low energies (u  1), the Klein-Nishina formula (2.10) is conveniently approximated by the expansion [34] σ (KN) =

 2 8π α hc ¯ 3 mc2

  Thomson cross section

  6 2 u 1 + 2u + + . . . , 5 (1 + 2u)2 1

2 The Interaction of Radiation with Matter

11

while at high energies (u  1) the approximation [8, 10, 22]  σ

(KN)

∼π

α hc ¯ mc2

2

  1 1 ln (2u) + u 2

can be used. The angular distribution of the scattered photon is given by the differential cross section dσ (KN) =π d (cos θ )

 

α hc ¯ mc2

2

× 1+ 

1 1 + u (1 − cos θ )

2 

1 + cos2 θ 2 



u2 (1 − cos θ )2  , 1 + cos2 θ [1 + u (1 − cos θ )]

which corresponds to a kinetic energy spectrum [22] dσ (KN) =π dT



α hc ¯ mc2

2

1 2 u mc2

 2+



T E−T

2

2 (E − T ) E−T 1 − + 2 u E uT



of the target electron. The cross section for Compton scattering off an atom scales roughly with the number of electrons in the atom and, assuming that the photon energy is large compared to the atomic binding energies, may be approximated by σ (Compton) ∼ Zσ (KN) . Methods for including the effects of the binding energy and the internal motion of the orbital electrons in calculations of atomic Compton scattering cross sections are discussed, for instance, in Ref. [35].

2.2.3 Pair Production For photon energies exceeding 2mc2, an interaction mechanism becomes possible where the incoming photon disappears and an electron-positron pair, with a total energy equal to the photon energy E, is created. Momentum conservation requires this process, which is closely related to bremsstrahlung (Sect. 2.4.1), to take place in the electric field of a nucleus or of the atomic electrons. In the latter case, kinematic constraints impose a threshold of E > 4mc2 .

12

H. Bichsel and H. Schindler

At high photon energies, the electron-positron pair is emitted preferentially in the forward direction and the absorption coefficient due to pair production can be approximated by μ = Nσ (pair production) =

7 1 , 9 X0

where X0 is a material-dependent parameter known as the radiation length (see Sect. 2.4.1). More accurate expressions are given in Ref. [8]. Tabulations of calculated pair-production cross sections can be found in Ref. [36] and are available online [24].

2.3 Interaction of Heavy Charged Particles with Matter The main ingredient for computing the energy loss of an incident charged particle due to interactions with the electrons of the target medium is the single-differential cross section with respect to the energy transfer E in a collision. In this section, we discuss the calculation of dσ/dE and its moments for “fast”, point-like particles. To be precise, we consider particles with a velocity that is large compared to the velocities of the atomic electrons, corresponding to the domain of validity of the first-order Born approximation. In the limit where the energy transfer E is large compared to the atomic binding energies, dσ/dE approaches the cross section for scattering off a free electron. For a spin-zero particle with charge ze and speed βc, the asymptotic cross section (per electron) towards large energy transfers is given by [8] 2 dσ 2πz2 (α hc) 1 ¯ = 2 2 dE mc β E2

 

    dσR 2 E 2 E 1−β 1−β = . Emax dE Emax

(2.11)

Rutherford cross section

Similar expressions have been derived for particles with spin 1 and spin 1/2 [8]. The maximum energy transfer is given by the kinematics of a head-on collision between a particle with mass M and an electron (mass m) at rest, Emax

  m  m 2 −1 + = 2mc β γ 1 + 2γ , M M 2 2 2

(2.12)

which for M  m becomes Emax ∼ 2mc2 β 2 γ 2 . These so-called “close” or “knock-on” collisions, in which the projectile interacts with a single atomic electron, contribute a significant fraction (roughly half) to the average energy loss of a charged particle in matter but are rare compared to “distant” collisions in which the particle interacts with the atom as a whole or with a group of

2 The Interaction of Radiation with Matter

13

atoms. For an accurate calculation of dσ/dE, the electronic structure of the target medium therefore needs to be taken into account. In the non-relativistic first-order Born approximation, the transition of an atom from its ground state to an excited state |n involving a momentum transfer q is characterised by the matrix element (inelastic form factor) Fn0 (q) = n|

Z 

 exp

j =1

 i q · rj |0, h¯

which is independent of the projectile. The differential cross section with respect to the recoil parameter Q = q 2 / (2m), derived by Bethe in 1930 [16], is given by [1–3, 16] 2 2 dσn 2πz2 (α hc) 1 fn (q) 2πz2 (α hc) ¯ ¯ 2 = |F = , (q)| n0 dQ mc2 β 2 Q2 mc2 β 2 QEn

where fn (q) denotes the generalised oscillator strength (GOS). In the limit q → 0 it becomes the dipole oscillator strength fn discussed in Sect. 2.2.1. The doubledifferential cross section for transitions to the continuum (i.e. ionisation) is given by 2 d2 σ 2πz2 (α hc) 1 df (E, q) ¯ = , dEdQ mc2 β 2 QE dE

(2.13)

where df (E, q) /dE is the generalised oscillator strength density. The GOS is constrained by the Bethe sum rule [2, 16] (a generalisation of the TRK sum rule), 

 fn (q) +

n

dE

df (E, q) = Z, dE

∀q.

(2.14)

Closed-form expressions for the generalised oscillator strength (density) exist only for very simple systems such as the hydrogen atom (Fig. 2.4). Numerical calculations are available for a number of atoms and molecules (see e.g. Ref. [37]). A prominent feature of the generalised oscillator strength density is the so-called “Bethe ridge”: at high momentum transfers df (E, q) /dE is concentrated along the free-electron dispersion relation Q = E. In order to calculate dσ/dE, we need to integrate the double-differential crosssection over Q, dσ = dE

Q max

dQ Qmin

d2 σ , dEdQ

Qmin ∼

E2 . 2mβ 2c2

(2.15)

14

H. Bichsel and H. Schindler

Fig. 2.4 Generalised oscillator strength density df (E, q) /dE of atomic hydrogen [2, 3, 16], for transitions to the continuum

For this purpose, it is often sufficient to use simplified models of the generalised oscillator strength density, based on the guidelines provided by model systems like the hydrogen atom, and using (measured) optical data in the low-Q regime. Equation (2.13) describes the interaction of a charged particle with an isolated atom, which is a suitable approximation for a dilute gas. In order to extend it to dense media and to incorporate relativistic effects, it is convenient to use a semiclassical formalism [19, 38]. In this approach, which can be shown to be equivalent to the first-order quantum mechanical result, the response of the medium to the incident particle is described in terms of the complex dielectric function.

2.3.1 Dielectric Theory Revisiting the energy loss of charged particles in matter from the viewpoint of classical electrodynamics, we calculate the electric field of a point charge ze moving with a constant velocity βc through an infinite, homogeneous and isotropic medium, that is we solve Maxwell’s equations ∇·B=0 ,

∇ ×E=−

1 ∂B , c ∂t

1 ∂D 4π + j, c ∂t c

∇ · D = 4πρ,

ρ = zeδ 3 (r − βct) ,

j = βcρ.

∇×B= for source terms

2 The Interaction of Radiation with Matter

15

The perturbation due to the moving charge is assumed to be weak enough such that there is a linear relationship between the Fourier components of the electric field E and the displacement field D, D (k, ω) = ε (k, ω) E (k, ω) , where ε (k, ω) = ε1 (k, ω) + iε2 (k, ω) is the (generalized) complex dielectric function. The particle experiences a force zeE (r = βct, t) that slows it down, and the stopping power is given by the component of this force parallel to the particle’s direction of motion, β dE = zeE · . dx β Adopting the Coulomb gauge k·A = 0, one obtains after integrating over the angles (assuming that the dielectric function ε is isotropic),   dE 2z2e2 =− 2 dω dk dx β π      

1 −1 ω2 ω 2 + ωk β − 2 2 Im . Im × kc2 ε (k, ω) k c −k 2 c2 + ε (k, ω) ω2 (2.16) The first term in the integrand represents the non-relativistic contribution to the energy loss which we would have obtained by considering only the scalar potential φ. It is often referred to as the longitudinal term. The second term, known as the transverse term, originates from the vector potential A and incorporates relativistic effects. On a microscopic level, the energy transfer from the particle to the target medium proceeds through discrete collisions with energy transfer E = hω ¯ and momentum transfer q = hk. Comparing Eq. (2.2) with the macroscopic result (2.16), one ¯ obtains d2 σ 2z2 α = 2 dEdq β π hcN ¯        1 −1 1 E2 1 2   × Im + β − 2 2 Im . q ε (q, E) q q c −1 + ε (q, E) E 2 / q 2 c2

(2.17)

16

H. Bichsel and H. Schindler

The loss function Im (−1/ε (q, E)) and the generalized oscillator strength density are related by   2Z −1 df (E, q) =E  . Im 2 dE ε (q, E) π h¯ p

(2.18)

Using this identity, we see that the longitudinal term (first term) in Eq. (2.17) is equivalent to the non-relativistic quantum mechanical result (2.13). As is the case with the generalized oscillator strength density, closed-form expressions for the dielectric loss function Im (−1/ε (q, E)) can only be derived for simple systems like the ideal Fermi gas [39, 40]. In the following (Sects. 2.3.2 and 2.3.3), we discuss two specific models of Im (−1/ε (q, E)) (or, equivalently, df (E, q) /dE).

2.3.2 Bethe-Fano Method The relativistic version of Eq. (2.13) or, in other words, the equivalent of Eq. (2.17) in oscillator strength parlance, is [1, 41] ⎡ d2 σ dEdQ

=

2πz2 (α hc)2 ¯ mc2 β 2

⎢ Z⎣

⎤

  Q ⎥ + 1 +  2    2 ⎦ 2 mc2 Q 1+ Q2 − E 2 Q2 1 + Q 2 |F (E, q)|2

|β t · G (E, q)|2

2mc

2mc

2mc

(2.19)   where Q 1 + Q/2mc2 = q 2 /2m, β t is the component of the velocity perpendicular to the momentum transfer q, and F (E, q) and G (E, q) represent the matrix elements for longitudinal and transverse excitations. Depending on the type of target and the range of momentum transfers involved, we can use Eqs. (2.13), (2.19) or (2.17) as a starting point for evaluating the singledifferential cross section. Following the approach described by Fano [1], we split dσ/dE in four parts. For small momentum transfers (Q < Q1 ∼ 1 Ry), we can use the non-relativistic expression (2.13) for the longitudinal term and approximate the generalised oscillator strength density by its dipole limit, 2 dσ (1) 1 df (E) 2πz2 (α hc) ¯ = dE mc2 β 2 E dE

Q1

2 dQ 1 df (E) Q1 2mc2 β 2 2πz2 (α hc) ¯ . = ln Q mc2 β 2 E dE E2

Qmin

(2.20) In terms of the dielectric loss function, one obtains   dσ (1) z2 α Q1 2mc2β 2 −1 = 2 ln . Im dE ε (E) β π hcN E2 ¯

2 The Interaction of Radiation with Matter

17

For high momentum transfers (Q > Q2 ∼ 30 keV), i.e. for close collisions where the binding energy of the atomic electrons can be neglected, the longitudinal and transverse matrix elements are strongly peaked at the Bethe ridge Q = E. Using [1]   1 + Q/ 2mc2  δ(E − Q),  |F (E, q)| ∼ 1 + Q/ mc2   2 2 2 1 + Q/ 2mc  δ(E − Q) |β t · G (E, q)| ∼ βt 1 + Q/ mc2 2

and βt2 =

  1  − 1 − β2  1 + Q/ 2mc2

one obtains (for longitudinal and transverse excitations combined), 2 2πz2 (α hc) Z dσ (h) ¯ = 2 2 dE mc β E

  E 1 − β2 . 1− 2mc2



(2.21)

In the intermediate range, Q1 < Q < Q2 , numerical calculations of the generalised oscillator strength density are used. An example of df (E, q) /dE is shown in Fig. 2.5. Since the limits Q1 , Q2 do not depend on the particle velocity, the integrals 2 2πz2 (α hc) 1 dσ (2) ¯ = dE mc2 β 2 E

Q2

dQ df (E, q) Q dE

Q1

need to be evaluated only once for each value of E. The transverse contribution can be neglected1 [41]. The last contribution, described in detail in Ref. [1], is due to low-Q transverse excitations in condensed matter. Setting Im (−1/ε (E, q)) = Im (−1/ε (E)) in the second term in Eq. (2.17) and integrating over q gives dσ (3) z2 α = 2 dE β πN hc ¯       −1 ε1 (E) π 1 1 − β 2 ε1 (E) 2   × Im ln − arctan . + β − 1 − β 2 ε (E) ε (E) 2 β 2 ε2 (E) |ε (E)|2

(2.22)

1 For

particle speeds β < 0.1, this approximation will cause errors, especially for M0 .

18

H. Bichsel and H. Schindler 0.12 0.10 df(E,q)/dE [Ry–1]

E=48 Ry 0.08 0.06 0.04 0.02 0.00

0

2

4

6

Kao

8

10

12

14

Fig. 2.5 Generalized oscillator strength density for Si for an energy transfer E = 48 Ry to the 2p-shell electrons [41–44], as function of ka0 (where k 2 a02 = Q/Ry). Solid line: calculated with Herman-Skilman potential, dashed line: hydrogenic approximation [45, 46]. The horizontal and vertical line define the FVP approximation (Sect. 2.3.3)

We will discuss this term in more detail in Sect. 2.3.3. The total single-differential cross section, dσ (1) dσ (2) dσ (3) dσ (h) dσ = + + + , dE dE dE dE dE is shown in Fig. 2.6 for particles with βγ = 4 in silicon which, at present, is the only material for which calculations based on the Bethe-Fano method are available.

2.3.3 Fermi Virtual-Photon (FVP) Method In the Bethe-Fano algorithm discussed in the previous section, the dielectric function ε (q, E) was approximated at low momentum transfer by its optical limit ε (E). In the Fermi virtual-photon (FVP) or Photoabsorption Ionisation (PAI) model [6, 47, 48], this approximation is extended to the entire domain q 2 < 2mE. Guided by the shape of the hydrogenic GOS, the remaining contribution to Im (−1/ε (q, E)) required to satisfy the Bethe sum rule 

∞ E Im 0

 2 π −1 dE = h

¯ p ε (q, E) 2

∀q,

(2.23)

2 The Interaction of Radiation with Matter

19

20.0

dσ/dE/dσR /dE

10.0 4.0 2.0 1.0 0.4 0.2 0.1

1

3

10

30

100

300

1000

3000 10000 30000 100000

E [eV]

Fig. 2.6 Differential cross section dσ/dE, divided by the Rutherford cross section dσR /dE, for particles with βγ = 4 in silicon, calculated with two methods. The abscissa is the energy loss E in a single collision. The Rutherford cross section is represented by the horizontal line at 1.0. The solid line was obtained [41] with the Bethe-Fano method (Sect. 2.3.2). The cross section calculated with the FVP method (Sect. 2.3.3) is shown by the dotted line. The functions all extend to Emax ∼ 16 MeV. The moments are M0 = 4 collisions/μm and M1 = 386 eV/ μm (Table 2.2)

is attributed to the scattering off free   electrons (close collisions). This term is thus of the form Cδ E − q 2 / (2m) , with the factor C being determined by the normalisation (2.23), 1 C= E

E



 −1 E Im dE  . ε (E  ) 

0

Combining the two terms, the longitudinal loss function becomes 

−1 Im ε (q, E)



    δ E− −1 q2 = Im  E− + ε (E) 2m E 

q2 2m



E

E  Im



 −1 dE  . ε (E  )

0

In the transverse √ term, the largest contribution to the integral comes from the region E ∼ qc/ ε, i.e. from the vicinity of the (real) photon dispersion relation, and one consequently approximates ε (q, E) by ε (E) throughout.

20

H. Bichsel and H. Schindler

10−1

10

E d /dE [Mb]

E d /dE [Mb]

10−1

βγ = 4 −2

β γ = 100 −2

10

10−3

10−3 10−4

10−4

10−5

10−5 102

103

104

105

102

E [eV]

103

105

104

E [eV]

Fig. 2.7 Differential cross section dσ/dE (scaled by the energy loss E) calculated using the FVP algorithm, for particles with βγ = 4 (left) and βγ = 100 (right) in argon (at atmospheric pressure, T = 20 ◦ C). The upper, unshaded area corresponds to the first term in Eq. (2.24), i.e. to the contribution from distant longitudinal collisions. The lower area corresponds to the contribution from close longitudinal collisions, given by the second term in Eq. (2.24). The intermediate area corresponds to the contribution from transverse collisions

The integration over q can then be carried out analytically and one obtains for the single-differential cross section dσ/dE ⎡ ⎤     E z2 α ⎣ −1 2mc2 β 2 1 −1 z2 α dσ  ⎦ = 2 Im ln + 2 dE + 2 E Im  dE β πN hc ε (E) E E ε (E ) β πN hc ¯ ¯  × Im

0







1 ε (E) −1  + β2 − 1 ln  1 − β 2 ε (E) ε (E) |ε (E)|2



π 1 − β 2 ε1 (E) − arctan 2 β 2 ε2 (E)



(2.24) The relative importance of the different terms in Eq. (2.24) is illustrated in Fig. 2.7. The first two terms describe the contributions from longitudinal distant and close collisions. The contribution from transverse collisions (third and fourth term) is identical to dσ (3) /dE in the Bethe-Fano algorithm. As can be seen from Fig. 2.7, its importance grows with increasing βγ . The third term incorporates the relativistic density effect, i.e. the screening of the electric field due to the polarisation of the medium induced by the passage of the charged particle. In the transparency region ε2 (E) = 0, the fourth term can be identified with √ the cross section for the emission of Cherenkov photons. It vanishes for β < 1/ ε; above this threshold it becomes   α dσ (C) 1 α = sin2 θC , 1− 2 ∼ dE N hc β ε N hc ¯ ¯

2 The Interaction of Radiation with Matter

21

where cos θC =

1 √ . β ε

Cherenkov detectors are discussed in detail in Chap. 7 of this book. In the formulation of the PAI model by Allison and Cobb [6], the imaginary part ε2 of the dielectric function is approximated by the photoabsorption cross section σγ , ε2 (E) ∼

N hc ¯ σγ (E) E

(2.25)

and the real part ε1 is calculated from the Kramers-Kronig relation 2 ε1 (E) − 1 = P π

∞ 0

  E  ε2 E  dE  . E 2 − E 2

In addition, the approximation |ε (E)|2 ∼ 1 is used. These are valid approximations if the refractive index2 is close to one (n ∼ 1) and the attenuation coefficient k is small. For gases, this requirement is usually fulfilled for energies above the ionisation threshold. Requiring only optical data as input, the FVP/PAI model is straightforward to implement in computer simulations. In the HEED program [49], the differential cross section dσ/dE is split into contributions from each atomic shell, which enables one to simulate not only the energy transfer from the projectile to the medium but also the subsequent atomic relaxation processes (Sect. 2.6). The GEANT4 implementation of the PAI model is described in Ref. [50]. For reasons of computational efficiency, the photoabsorption cross section σγ (E) is parameterised as a fourth-order polynomial in 1/E. FVP calculations for Ne and Ar/CH4 (90:10) are discussed in Ref. [51].

2.3.4 Integral Quantities For validating and comparing calculations of the differential cross section, it is instructive to consider the moments Mi of Ndσ/dE, in particular the inverse mean free path M0 and the stopping power M1 .

2 The

complex refractive index and the dielectric function are related by n + ik =

√

ε.

22

H. Bichsel and H. Schindler

2.3.4.1 Inverse Mean Free Path In the relativistic first-order Born approximation, the inverse mean free path for ionising collisions has the form [1, 2] M0 =

      2 2πz2 (α hc) ¯ 2 2 2 2 ln β − β + C , N M γ mc2 β 2

(2.26)

where  M = 2

1 df (E) dE, E dE

  4 C = M ln c˜ + ln 2 , α 2

and c˜ is a material-dependent parameter that can be calculated from the generalised oscillator strength density. Calculations can be found, for example, in Refs. [53, 54]. As in the Bethe stopping formula (2.28) discussed below, a correction term can be added to Eq. (2.26) to account for the density effect [55]. The inverse mean free path for dipole-allowed discrete excitations is given by [2] (n) M0

   2 2πz2 (α hc) fn 4 ¯ 2 2 2 = N ln β γ − β + ln c˜n + ln 2 . mc2 β 2 En α

We can thus obtain a rough estimate of the relative frequencies of excitations  and ionising collisions from optical data. In argon, for instance, the ratio of fn /En and M 2 is ∼20% [25]. For gases, M0 can be determined experimentally by measuring the inefficiency of a gas-filled counter operated at high gain (“zero-counting method”). Results (in the form of fit parameters M 2 , C) from an extensive series of measurements, using electrons with kinetic energies between 0.1 and 2.7 MeV, are reported in Ref. [52]. Other sets of experimental data obtained using the same technique can be found in Refs. [56, 57]. Table 2.1 shows a comparison between measured and calculated values (using the FVP algorithm) of M0 for particles with βγ = 3.5 at a temperature of 20 ◦ C and atmospheric pressure. The inverse mean free path is Table 2.1 Measurements [52] and calculations (using the FVP algorithm as implemented in HEED [49]) of M0 for βγ = 3.5 at T = 20 ◦ C and atmospheric pressure

Gas Ne Ar Kr Xe CO2 CF4 CH4 iC4 H10

M0 [cm−1 ] Measurement 10.8 23.0 31.5 43.2 34.0 50.9 24.6 83.4

FVP 10.5 25.4 31.0 42.1 34.0 51.8 29.4 90.9

23

M0 [μm-1]

M1 [keV/μm]

2 The Interaction of Radiation with Matter

1

10

1

102

10

103

1

βγ

10

102

103

βγ

Fig. 2.8 Inverse ionisation mean free path (left) and stopping power (right) of heavy charged particles in silicon as a function of βγ , calculated using the Bethe-Fano algorithm (solid line) and the FVP model (dashed line). The two stopping power curves are virtually identical

sensitive to the detailed shape of the differential cross section dσ/dE at low energies and, consequently, to the optical data used. Figure 2.8(left) shows M0 in solid silicon as a function of βγ , calculated using the Bethe-Fano and FVP algorithms. The difference between the results is ∼6 − 8%, as can also be seen from Table 2.2. Owing to the more detailed (and more realistic) modelling of the generalised oscillator strength density at intermediate Q, the Bethe-Fano algorithm can be expected to be more accurate than the FVP method.

2.3.4.2 Stopping Power Let us first consider the average energy loss of a non-relativistic charged particle in a dilute gas, with the double-differential cross section given by Eq. (2.13), 2 2πz2 (α hc) dE ¯ = − N dx mc2 β 2

E max

Q max

dE Emin

dQ df (E, q) . Q dE

Qmin

As an approximation, we assume that the integrations over Q and E can be interchanged and the integration limits Qmin , Q  max (which depend on E) be replaced by average values Qmin = I 2 / 2mβ 2 c2 , Qmax = Emax [58]. Using the Bethe sum rule (2.23), we then obtain −

2 2πz2 (α hc) dE 2mc2β 2 Emax ¯ = NZ ln , dx mc2 β 2 I2

24

H. Bichsel and H. Schindler

where the target medium is characterised by a single parameter: the “mean ionisation energy” I , defined by ln I =

1 Z

 dE ln E

df (E) dE

in terms of the dipole oscillator strength density, or ln I =

2

2  π h

¯ p



 dE E Im

 −1 ln E. ε (E)

(2.27)

in terms of the dielectric loss function. In the relativistic case, one finds the well-known Bethe stopping formula −

 2 2πz2 (α hc) dE 2mc2 β 2 γ 2 Emax ¯ 2 = NZ ln − 2β − δ , dx mc2 β 2 I2

(2.28)

where δ is a correction term accounting for the density effect [59]. Sets of stopping power tables for protons and alpha particles are available in ICRU report 49 [60] and in the PSTAR and ASTAR online databases [61]. Tables for muons are given in Ref. [62]. These tabulations include stopping power contributions beyond the first-order Born approximation, such as shell corrections [42, 45, 46] and the Barkas-Andersen effect [63–65]. The stopping power in silicon obtained from the Bethe-Fano algorithm (Sect. 2.3.2) has been found to agree with measurements within ±0.5% [41]. As can be seen from Table 2.2 and Fig. 2.8, FVP and Bethe-Fano calculations for M1 in silicon are in close agreement, with differences Ecut is given by −1

λ

T = M0 = N Ecut

dσrad dE. dE

2 The Interaction of Radiation with Matter

27

If we neglect the term (1 − u) /9 in Eq. (2.31), we find for the radiative stopping power at T  mc2 dE = M1 = N − dx

T E

T dσrad dE ∼ , dE X0

(2.32)

183 , Z 1/3

(2.33)

0

where the parameter X0 , defined by 1 = 4α 3 X0



h¯ c mc2

2 NZ (Z + 1) ln

is known as the radiation length. Values of X0 for many commonly used materials can be found in Ref. [70] and on the PDG webpage [71]. Silicon, for instance, has a radiation length of X0 ∼ 9.37 cm [71]. Being approximately proportional to the kinetic energy of the projectile, the radiative stopping power as a function of T increases faster than the average energy loss due to ionising collisions given by Eq. (2.28). At high energies— more precisely, above a so-called critical energy (∼38 MeV in case of silicon [71])—bremsstrahlung therefore represents the dominant energy loss mechanism of electrons and positrons.

2.5 Energy Losses Along Tracks: Multiple Collisions and Spectra Consider an initially monoenergetic beam of identical particles traversing a layer of material of thickness x. Due to the randomness both in the number of collisions and in the energy loss in each of the collisions, the total energy loss  in the absorber will vary from particle to particle. Depending on the use case, the kinetic energy of the particles, and the thickness x, different techniques for calculating the probability distribution f (, x)—known as “straggling function” [72]—can be used. Our focus in this section is on scenarios where the average energy loss in the absorber is small compared to the kinetic energy T of the incident particle (as is usually the case in vertex and tracking detectors), such that the differential cross section dσ/dE and its moments do not change significantly between the particle’s entry and exit points in the absorber. The number of collisions n then follows a Poisson distribution p (n, x) =

nn −n e , n!

(2.34)

with mean n = xM0 . The probability f (1) (E) dE for a particle to lose an amount of energy between E and E + dE in a single collision is given by the normalised

28

H. Bichsel and H. Schindler

differential cross section, f (1) (E) =

dσ 1 , N M0 dE

and the probability distribution for a total energy loss  in n collisions is obtained from n-fold convolution of f (1) ,    f (n) () = f (1) ⊗ f (1) ⊗ · · · ⊗ f (1) () = dE f (n−1) ( − E) f (1) (E) ,

  n times

as illustrated in Figs. 2.9 and 2.10. The probability distribution for a particle to suffer a total energy loss  over a fixed distance x is given by [72, 73] f (, x) =

∞ 

p (n, x) f (n) () ,

(2.35)

n=0

where f (0) () = δ (). Equation (2.35) can be evaluated in a stochastic manner (Sect. 2.5.1), by means of direct numerical integration (Sect. 2.5.2), or by using integral transforms (Sect. 2.5.3). 0.08 n=2

n=3

0.06

*n

s [E] ù f (n) [a.u]

n=1

0.004

0.002

0.00

10

20

30 E [eV]

50

70

100

Fig. 2.9 Distributions f (n) of the energy loss in n collisions (n-fold convolution of the singlecollision energy loss spectrum) for Ar/CH4 (90:10)

2 The Interaction of Radiation with Matter

0.100

n=1

29

n=2

*n

s [E] ù f (n) [a.u]

0.050

n=3 n=4

0.020

n=5

0.010 0.005

0.002 0.001

0

20

40

60

80

100

E [eV]

Fig. 2.10 Distributions f (n) of the energy loss in n collisions for solid silicon. The plasmon √ peak at ∼17 eV appears in each spectrum at E ∼ n × 17 eV, and its FWHM is proportional to n. The structure at ∼2 eV appears at 2 + 17(n − 1) eV, but diminishes with increasing n. For n = 6 (not shown) the plasmon peak (at 102 eV) merges with the L-shell energy losses at 100 eV, also see Fig. 2.12

2.5.1 Monte Carlo Method In a detailed Monte Carlo simulation, the trajectory of a single incident particle is followed from collision to collision. The required ingredients are the inverse mean (i) free path M0 and the cumulative distribution function,



(i)

(E) =

E

1

N

(i)

M0

dσ (i)  dE , dE 

(2.36)

0

for each interaction process i (electronic collisions, bremsstrahlung, etc.) to be taken into account in the simulation. The distance x between successive collisions follows an exponential distribution and is sampled according to x = −

ln r , λ−1

where r ∈ (0, 1] is a uniformly distributed random number and λ−1 =

 i

(i)

M0

30

H. Bichsel and H. Schindler

is the total inverse mean free path. After updating the coordinates of the particle, the collision mechanism to take place is chosen based on the relative frequencies (i) M0 /λ−1 . The energy loss in the collision is then sampled by drawing another uniform random variate u ∈ [0, 1], and determining the corresponding energy loss E from the inverse of the cumulative distribution, E = −1 (u) . In general, the new direction after the collision will also have to be sampled from a suitable distribution. The above procedure is repeated until the particle has left the absorber. The spectrum f (, x) is found by simulating a large number of particles and recording the energy loss  in a histogram. Advantages offered by the Monte Carlo approach include its straightforward implementation, the possibility of including interaction mechanisms other than inelastic scattering (bremsstrahlung, elastic scattering etc.), and the fact that it does not require approximations to the shape of dσ/dE to be made. For thick absorbers, detailed simulations can become unpractical due to the large number of collisions, and the need to update the inverse mean free path M0 and the cumulative distribution  (E) following the change in velocity of the particle. In “mixed” simulation schemes, a distinction is made between “hard” collisions which are simulated individually, and “soft” collisions (e.g. elastic collisions with a small angular deflection of the projectile, or emission of low-energy bremsstrahlung photons) the cumulative effect of which is taken into account after each hard scattering event. Details on the implementation of mixed Monte Carlo simulations can be found, for example, in the PENELOPE user guide [74].

2.5.2 Convolutions For short track segments, one can calculate the distributions f (n) explicitly by numerical integration and construct f (, x) directly using Eq. (2.35). A computationally more efficient approach is the absorber doubling method [41, 75], which proceeds as follows. Consider a step x that is small compared to the mean free path such that n  1 (in practice: n < 0.01 [76]). Expanding Eq. (2.35) in powers of n and retaining only constant and linear terms gives f (E, x) ∼ (1 − n) f (0) (E) + nf (1) (E) . The straggling function for a distance 2x is then calculated using  f ( − E, x) f (E, x) dE.

f (, 2x) = 0

2 The Interaction of Radiation with Matter

31

This procedure is carried out k times until the desired thickness 2k x is reached. Because of the tail of f (1) (E) towards large energy transfers, the numerical convolution is performed on a logarithmic grid. More details of the implementation can be found in Refs. [75, 76].

2.5.3 Laplace Transforms In the Laplace domain, Eq. (2.35) becomes F (s, x) = L{f (, x)} = e−n

∞  nn n=0

n!

⎡

= exp ⎣−Nx

∞

L{f (1) ()}n ⎤  dσ  ⎦. dE 1 − e−sE dE

0

Following Landau [20], we split the integral in the exponent in two parts, ∞



dE 1 − e

Nx

1  ∞   dσ   dσ −sE dσ = Nx dE 1 − e + Nx dE 1 − e−sE , dE dE dE E

−sE

0

0

E1

where E1 is chosen to be large compared to the ionisation threshold while at the same time satisfying sE1  1. For energy transfers exceeding E1 , the differential cross section is assumed to be given by the asymptotic expression for close collisions (2.11); for E < E1 , it is not specified. Using exp (−sE) ∼ 1 − sE, we obtain for the first term E1 I1 = Nx

E1  dσ  dσ −sE ∼ Nxs dE 1−e E. dE dE dE

0

0

We can therefore evaluate I1 by subtracting the contribution due to energy transfers between E1 and Emax according to Eq. (2.11) from the total average energy loss xdE/dx = ,   Emax − β2 , I1 ∼ s − sξ ln E1 where we have introduced the variable ξ =x

2 NZ 2πz2 (α hc) ¯ . 2 2 mc β

32

H. Bichsel and H. Schindler

For evaluating the second integral, we approximate dσ/dE by the Rutherford cross section dσR /dE ∝ 1/E 2 . Because of the rapid convergence of the integral for sE  1, we further assume that the upper integration limit can be extended to infinity (instead of truncating dσ/dE at Emax ). Integrating by parts and substituting z = sE yields ∞ I2 = ξ

dE

1 − e−sE

=ξ

E2

E1

e−sE1

1− E 1 ∼s



⎛

∞ +ξ s

dz

e−z z

⎜ ∼ ξ s ⎝1 +

sE1

1

⎞ dz ⎟ − C⎠ , z

sE1

where C ∼ 0.577215665 is Euler’s constant.3 Combining the two terms I1 and I2 , one obtains

   − ln sEmax + β 2 , F (s, x) = exp −ξ s 1 − C + ξ and, applying the inverse Laplace transform, f (, x) = L−1 {F (s, x)} =

1 φL (λ) , ξ

(2.37)

where c+i∞ 

1 φL (λ) = 2πi

du eu ln u+λu .

(2.38)

c−i∞

is a universal function of the dimensionless variable  −  ξ − (1 − C) − β 2 − ln . ξ Emax

λ=

The maximum of φL (λ) is located at λ ∼ −0.222782 and the most probable energy loss is, consequently, given by   p ∼  + ξ 0.2 + β 2 + ln κ ,

3

1 −C =

dz 0

e−z − 1 + z

∞ dz 1

e−z = −0.577215665 . . . z

(2.39)

2 The Interaction of Radiation with Matter

33

where κ = ξ/Emax . The full width at half maximum (FWHM) of the Landau distribution4 (2.37) is approximately 4.02ξ . A somewhat unsatisfactory aspect of φL (λ) is that its mean is undefined (a consequence of allowing arbitrarily large energy transfers E > Emax ). This deficiency was overcome by Vavilov [77] who, taking account of the kinematically allowed maximum energy transfer Emax and using the differential cross section (2.11) in I2 , obtained 1 f (, x) = φV (λ) , ξ

1 κ 1+β 2 C  e φV (λ) = 2πi

c+i∞ 

exp (ψ (u) + λu) du,

c−i∞

where 



⎛

⎜ ψ (u) = u ln κ + u + β 2 κ ⎝

∞

⎞ e−t

u⎟ dt + ln ⎠ − κe−u/κ . t κ

u/κ

For small values of κ (κ < 0.01 [77]) the Vavilov distribution tends to the Landau distribution, while   for κ  1 it approaches a Gaussian distribution with σ 2 = ξ Emax 1 − β 2 /2 [78]. Algorithms for the numerical evaluation of φL and φV and for drawing random numbers from these distributions are discussed e.g. in Refs. [78–81] and are implemented in ROOT [82]. Attempts have been made [83, 84] to improve the Landau-Vavilov method with respect to the treatment of distant collisions by including the second order term in the expansion of exp (−sE) in I1 . The results are akin to convolving φL or φV with a Gaussian distribution (expressions for estimating the standard deviation σ of the Gaussian are reviewed in Ref. [41]).

2.5.4 Examples Let us first consider track segments for which the projectile suffers on average only tens of collisions. At the minimum of M0 , n = 10 corresponds to a track length x ∼ 4 mm for argon (at atmospheric pressure, T = 20 ◦C) and x ∼ 2 μm for silicon (Tables 2.1 and 2.2). As can be seen from Figs. 2.11 and 2.12, the features of the differential cross section dσ/dE are clearly visible in the straggling functions f (, x). These spectra cannot be described by a Landau distribution (or variants thereof) and need to be calculated using Monte Carlo simulation or numerical convolution.

4 In high-energy physics parlance, the term “Landau distribution” is sometimes used for energy loss spectra f (, x) in general. In this chapter, it refers only to the distribution given by Eq. (2.37).

34

H. Bichsel and H. Schindler

d

1.0

f [Δ]

0.8

c 1 mm 5 mm

0.6

4 mm

3 mm

0.4

2 mm 0.2

0.0 10

20

50

100

Δ [eV]

200

500

1000

2000

Fig. 2.11 Straggling functions for singly charged particles with βγ = 4.48 traversing segments of length x = 1 . . . 5 mm in Ar. The inverse mean free path M0 is 30 collisions/cm. The functions are normalised to unity at the most probable value. The broad peak at ∼17 eV is due to single collisions, see Fig. 2.9. For two collisions it broadens and shifts to about 43 eV, marked c, and for n = 3 it can be seen at d. It may be noted that the peak at 11.7 eV (if the function is normalised to unit area) is exactly proportional to n exp (−n), as expected from Eq. (2.35). Energy losses to L-shell electrons of Ar (with a binding energy of ∼250 eV) appear at e, for x = 1 mm they have an amplitude of 0.04. For x > 2 mm, peak c disappears, and peak d becomes the dominant contribution defining the most probable energy loss p . The buildup for peak e at 440–640 eV is the contribution from L-shell collisions. It appears roughly at 250 eV+p . The inverse mean free path for collisions with E > 250 eV is only 1.7 collisions/cm, thus the amplitude of the peak e is roughly proportional to x. The Bethe mean energy loss is 250 eV/mm

With increasing number of collisions, the detailed features of the differential cross section become “washed out” and the energy loss spectra f (, x) tend to the Landau shape but are typically broader, as shown in Figs. 2.13 and 2.14. Reasonable agreement with measured energy loss spectra for thin absorbers can often be achieved by fit functions based on the convolution of a Landau/Vavilov distribution and a Gaussian distribution. For a predictive calculation of f (, x), however, numerical convolution or a Monte Carlo simulation are usually needed.

2.5.5 Methods for Thick Absorbers In order to compute the energy loss distribution for a layer of material in which the kinetic energies T of the traversing particles change considerably (i.e. by more than 5–10% [86]), we divide the absorber in segments of length x that are sufficiently small such that the straggling function f (, x) can be calculated using the methods for thin absorbers described above. Let φ (y, T ) be the distribution of

2 The Interaction of Radiation with Matter

35

100

2 GeV protons 1 μm Si

f [Δ]

80

60

40

20

0

0

50

100

150 Δ [eV]

200

250

300

Fig. 2.12 Straggling in 1 μm of Si (n = 4) for particles with βγ = 2.1, compared to the Landau function (dashed line). The Bethe mean energy loss is  = 400 eV. Measured straggling functions of this type are given in Ref. [85]

0.6

f [Δ] [1/kev]

0.4 w 0.2

Δp

0.0

1

2

Δ [keV]

3

4

5

Fig. 2.13 Straggling function f () for particles with βγ = 3.6 traversing 1.2 cm of Ar gas (n = 36) calculated using the convolution method (solid line) compared to the Landau distribution (dashed line). Parameters describing f () are the most probable energy loss p , i.e. the position of the maximum of the straggling function, at 1371 eV, and the full width at half maximum (FWHM) w = 1463 eV. The Bethe mean energy loss is  = 3044 eV. The peak of the Landau function is at 1530 eV

36

H. Bichsel and H. Schindler

w / Δp

Fig. 2.14 Relative width (full width at half maximum w divided by the most probable value p ) of the straggling spectrum f (, x) as function of the absorber thickness x, for particles with βγ ∼ 3.16 in silicon. The dashed line corresponds to the relative width of the Landau distribution. Circles represent results of a Monte Carlo simulation using the Bethe-Fano differential cross section

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

102

103 x [μm]

kinetic energies at a distance y in the absorber. If f (, x) is known for all T , the spectrum of kinetic energies at y + x can be calculated using  φ (y + x, T ) =

φ (y, T + ) f (, x; T + ) d.

Scaling relations, discussed in Ref. [51], can be used to limit the number of thinabsorber distributions f (, x; T ) that need to be tabulated. In a “condensed history” Monte Carlo simulation [87], the energy loss spectrum is calculated stochastically by sampling the energy loss over a substep x from a suitable thin-absorber distribution (e.g. a Vavilov function), and updating the kinetic energy T of the projectile after each substep.

2.6 Energy Deposition Leaving the emission of Cherenkov radiation and other collective effects aside, charged-particle collisions with electrons in matter result in the promotion of one of the electrons in the target medium to a bound excited state or to the continuum. Both effects (excitation and ionisation) can be exploited for particle detection purposes. In scintillators, discussed in Chap. 3 of this book, part of the energy transferred to excitations is converted to light. Detectors based on ionisation measurement in gases and semiconductors are discussed in Chaps. 4 and 5. In the following we briefly review the main mechanisms determining the number of electron-ion pairs (in gases) or electron-hole pairs (in semiconductors) produced in the course of an ionising collision, along with their spatial distribution.

2 The Interaction of Radiation with Matter

37

V L3 L2 L1

K

Fig. 2.15 After the ejection of an inner-shell electron, the resulting vacancy is filled by an electron from a higher shell. The energy released in the transition can either be carried away by a fluorescence photon (left) or be transferred to an electron in a higher shell (Auger process, middle). Coster-Kronig transitions (right) are Auger processes in which the initial vacancy is filled by an electron from the same shell

2.6.1 Atomic Relaxation If a charged-particle collision (or a photoabsorption interaction) ejects an innershell electron from an atom, the resulting vacancy will subsequently be filled by an electron from a higher shell, giving rise to a relaxation chain which can proceed either radiatively, i.e. by emission of a fluorescence photon, or radiation-less (Auger effect). The two processes are illustrated schematically in Fig. 2.15. Fluorescence photons can in turn ionise another atom in the medium or, with a probability depending on the geometry of the device, escape from the detector. The fluorescence yield, i.e. the probability for a vacancy to be filled radiatively, increases with the atomic number Z: in silicon, for example, the average fluorescence yield is ∼5%, compared to ∼54% in germanium [88]. Compilations of fluorescence yields can be found in Refs. [88–91]. Tabulations of transition probabilities are available in the EADL database [92, 93].

2.6.2 Ionisation Statistics The “primary” ionisation electron knocked out in a collision (and also the Auger electrons) may have kinetic energies exceeding the ionisation threshold of the medium and thus undergo further ionising collisions along their path. Electrons with a kinetic energy T that is large compared to the ionisation threshold are referred to as “delta” electrons; their energy distribution follows approximately the close-collision differential cross section, given by Eq. (2.11) for spin-zero particles. The number of electrons ne produced in the energy degradation cascade of a delta electron with initial kinetic energy T is subject to fluctuations. The mean and variance of the

38

H. Bichsel and H. Schindler

distribution of ne are described by the average energy W required to produce an electron-ion (electron-hole) pair, ne  =

T , W

(2.40)

and the Fano factor F [94], σ 2 = (n − n)2  = F ne ,

(2.41)

respectively. Both W and F are largely determined by the relative importance of ionising and non-ionising inelastic collisions, the latter including e.g. excitations or phonon scattering. If the cross sections for these processes are known, the distribution of ne can be calculated using detailed Monte Carlo simulations. An example is the MAGBOLTZ program [95, 96], which includes the relevant cross sections for many commonly used detection gases. Inelastic cross sections of delta electrons in solids can be calculated based on the dielectric formalism discussed in Sect. 2.3.1 (in its non-relativistic version), often making using of optical data and a suitable model of the q-dependence of Im (−1/ε (q, E)) as, for instance, in the Penn algorithm described in Ref. [97]. Measurements of W for electrons in gases as a function of the electron’s initial kinetic energy are reported in Refs. [98–100, 102, 103]. As can be seen from Fig. 2.16, which shows measurements and calculations for CO2 , W increases towards low kinetic energies, while in the keV range and above it depends only weakly on T . For most gases and semiconductors typically used as sensitive media in particle detectors, the asymptotic (high-energy) W values are fairly well established. A compilation of recommended average W values, based on experimental data until 1978, is given in ICRU report 31 [101]. Critical reviews of W values and Fano factors including also more recent data can be found in Ref. [104]

100

W [eV]

Fig. 2.16 W value for electrons in CO2 as a function of the electron’s initial kinetic energy according to measurements by Combecher [98] (circles), Smith and Booz [99] (triangles), and Waibel and Grosswendt [100] (squares). The grey band represents results of a Monte Carlo calculation using the cross sections implemented in M AGBOLTZ [96]. The hatched band corresponds to the high-energy value recommended in Ref. [101]

90 80 70 60 50 40 30 20

102

104 103 kinetic energy [eV]

2 The Interaction of Radiation with Matter Table 2.3 Asymptotic W values and Fano factors for different gases and for solid silicon (at T = 300 K)

39

Ne Ar Kr Xe CO2 CH4 iC4 H10 CF4 Si

W [eV] 35.4 ± 0.9 [101] 26.4 ± 0.5 [101] 24.4 ± 0.3 [101] 22.1 ± 0.1 [101] 33.0 ± 0.7 [101] 27.3 ± 0.3 [101] 23.4 ± 0.4 [101] 34.3 [107] 3.67 ± 0.02 [108]

F 0.13–0.17 [104] 0.15–0.17 [104] 0.17–0.21 [104] 0.124–0.24 [104] 0.32 [104] 0.22–0.26 [104] 0.261 [106] > kT, or the radiative decay τ γ E0 ) − y/ β 2 E0 /cm,

(4.5)

with E0 in keV, and y = 0.114 for Ar, and y = 0.064 for Ne [20]; β= particle velocity/speed of light in vacuum. Thus, in Ar, for β~1 and E0 = 10 keV, P = 0.011/cm, i.e., on average one collision with E > 10 keV will occur on a track length of 90 cm. The range of a 10 keV electron is about 1.4 mm. The range decreases very rapidly with decreasing energy and is only about 30 μm for a 1 keV electron. It tums out that, although the majority of the ‘clusters’ consist of a single electron, clusters of size >1 contribute significantly to the mean total number nT of electrons produced per cm, so that nT is significantly larger than np , the mean number of primary clusters per cm. Table 4.1 gives experimental values for some of the common detector gases. In Ar at NTP one finds on average np = 26 and nT = 100 electrons/cm for a minimum ionizing particle, where nT depends on the volume around the track taken into account. The most probable value for the total ionization is nmp = 42 electrons/cm. The big difference between nT and nmp is an indication of the long tail of the distribution. ◦

Table 4.1 Properties of gases at 20 C, 1 atm Gas He Ne Ar Xe CH4 CO2 i-butane CF4

np 4.8 13 25 41 37 35 90 63

nT 7.8 50 100 312 54 100 220 120

w [eV] 45 30 26 22 30 34 26 54

EI [eV] 24.5 21.6 15.7 12.1 12.6 13.8 10.6 16.0

Ex [eV] 19.8 16.7 11.6 8.4 8.8 7.0 6.5 10.0

p [mg/cm3 ] 0.166 0.84 1.66 5.50 0.67 1.84 2.49 3.78

np , nT mean primary and total number of electron-ion pairs per cm; w: average energy dissipated per ion pair; EI , Ex : lowest ionization and excitation energy [21]

4 Gaseous Detectors

95

4.2.1.2 Cluster Size Distribution The space resolution in gaseous detectors is influenced not only by the Poisson distribution of the primary clusters along the track but also by the cluster size distribution, i.e., by the number of electrons in each cluster and their spatial extent. Little was known experimentally (except for some measurements in cloud chambers) until the first detailed theoretical study [22] for Ar at 1 atm and ◦ 20 C. Based on the experimental cross-sections for photo absorption, the oscillator strengths and the complex dielectric constants are calculated and from this the distribution of energy transfers larger than the ionization energy (15.7 eV). Finally, a detailed list is obtained for the distribution of cluster sizes for γ = 4 and γ = 1000, to estimate the relativistic rise. A cut of 15 keV was applied to the maximum energy transfer, thus concentrating on the local energy deposition. The mean number of clusters is found to be np = 26.6/cm at γ = 4, and 35/cm at γ = 1000. For a MIP, 80.2% of the clusters are found to contain 1 electron, 7.7% two electrons, 2% three electrons, and 1.4% more than 20 electrons. Several years later, a detailed experimental study of several gases is reported in [23]. For Ar, 66/15/6 and 1.1% of the clusters are found to contain 1/2/3 and ≥20 electrons, respectively. The values for low cluster sizes are quite different from the calculated values mentioned above and the calculated bump around 10 electrons is not seen in the measurement, see Fig. 4.1. The authors suggest as a possible explanation that one assumption made in the simulation may not be appropriate, namely that the absorption of virtual photons can be treated like that of real photons, which also leads to the bump at the L-absorption edge. A simpler model starting from measured spectra of electrons ejected in ionizing collisions gives good agreement with the measurements, in particular for the probability of small cluster sizes.

Fig. 4.1 Cluster size distribution: simulation for Ar (continuous line) [22] and measurements in Ar/CH4 (90/10%) [23]

96

H. J. Hilke and W. Riegler

Space resolution in drift chambers is influenced by the clustering in several ways. The arrival time of the first n electrons, where n times gas amplification is the threshold for the electronics, depends both on the spatial distribution of the clusters and the cluster size. For large clusters, δ-electrons, ionization may extend far off the trajectory.

4.2.1.3 Total Number of Ion Pairs The detector response is related to the cluster statistics but also to the total ionization nT , e.g., in energy measurements. A quantity W has been introduced to denote the average energy lost by the ionizing particle for the creation of one ion pair: W = Ei /nE ,

(4.6)

where Ei is the initial kinetic energy and nE the average total number of ion pairs after full dissipation of Ej . Measurements of W by total absorption of low energy particles show that it is practically independent of energy above a few keV for electrons and above a few MeV for α-particles. For that reason the differential value w, defined by w = x < dE/dx > / < nT >

(4.7)

may be used alternatively, as is usually done in Particle Physics, to relate the average total number of ion pairs nT , created in the track segment of length x, to the average energy lost by the ionizing particle. For relativistic particles, dE/dx can not be obtained directly from the difference of initial and final energy (about 270 keV/m for γ = 4 in Ar), as it is below the measurement resolution. Therefore, w has to be extrapolated from measurements of lower energy particles. For the rare gases one finds w/I = 1.7 − 1.8 and for common molecular gases w/I = 2.1 − 2.5, where I is the ionization potential, indicating the significant fraction of dE/dx spent on excitation. Values for photons and electrons are the same, also for α particles in rare gases; in some organic vapours they may be up to 15% higher for α-particles. At low energy, close to the ionization potential, W increases. In gas mixtures, where an excitation level of component A is higher than I of component B, excited molecules of A often produce a substantial increase in ionization, as has e.g. been observed even with minute impurities in He and Ne: adding 0.13% of Ar to He changed W from 41.3 to 29.7 eV per ion pair. This energy transfer is called Jesse effect or Penning effect, if metastable states are involved. It is also possible that more than one electron is ejected from a single atom, e.g., by Auger effect following inner shell ionization. The distribution of nT in small gas segments is very broad, see an example in Fig. 4.2 [24]. To describe the measurement result, it is thus appropriate to use the most probable value instead of the mean, since the mean of a small number of measurements will depend strongly on some events from the long tail

4 Gaseous Detectors

97

Fig. 4.2 Measured pulse height distribution for 2.3 cm in Ar/CH4 at 1 atm: (a) protons 3 GeV/c, (b) electrons 2 GeV/c [24]

of the distribution. The measured pulse height spectrum contains some additional broadening from the fluctuations of the avalanche process. For a mixture of Ar and 5% CH4 , a most probable value of nmp = 48 ion pairs/cm was found for minimum ionizing particles [25].

4.2.1.4 Dependence of Energy Deposit on Particle Velocity As mentioned above, for position detectors one is interested in the ionization deposited close to the particle trajectory. The Bethe-Bloch formula for dE/dx describes instead the average total energy loss from the incoming particle, including the energy spent on the ejection of energetic δ-electrons which deposit ionization far from the trajectory. To describe the local energy deposit, it is sensible to exclude the contribution from these energetic δ-electrons. This is done by replacing the maximum possible energy transfer Tmax by a cut-off energy Tcut > (3/2)kT, which is often fulfilled for drift of electrons in particle detectors, one obtains √ u2 = (eE/mNσ ) (Δ/2) , and √ c2 = (eE/mNσ ) (2/Δ) for ε − εE >> (3/2) kT .

(4.16)

100

H. J. Hilke and W. Riegler

Fig. 4.3 Electron collision cross-sections for Argon and Methane used in Magboltz [13, 27]

Fig. 4.4 The fraction  of energy lost per collision as function of mean energy ε of the electron [28]

The rigorous theory assuming a Dryvestem distribution for the random velocities c adds a multiplication factor of 0.85 to the right sides. It is important to note that E and N only appear as E/N, the reduced electric field, for which often a special unit is used: one Townsend (Td) with 1 Td =10−17 Vcm2 . The important role of σ and  is obvious; both depend on ε. Below the first excitation level the scattering is elastic and  − 2m/M − 10−4 for electrons scattered on gas molecules with mass M. For a high drift speed a small σ is required. Figure 4.3 shows the cross-sections σ for Ar and CH4 . A pronounced minimum, the so-called ‘Ramsauer dip’ is clearly visible. It leads to high drift velocities in Ar CH4 mixtures at low E-values. TPCs take advantage of this. ◦ 10−17 Vcm2 =250 V/cm atm at 20 C. From precise measurements of drift velocity u (to 1%) and longitudinal diffusion D/μ (to 3–5%), σ and  have been deduced for some gases [28]. The consistency of the calculated values with measurements of u and D/μ in various other gas mixtures gives confidence in the method. Figure 4.4 presents calculated values for  as function of ε. Figure 4.5 shows εk = (2/3)ε derived in the same way in another

4 Gaseous Detectors

101

Fig. 4.5 Values for the electron energy ε derived from diffusion measurements as function of the reduced electrical field [29]

Fig. 4.6 Some examples measured electron drift velocities. (left) [30], (rights) [31]

study for two extremes, cold CO2 and hot Ar [29]. Gases are denoted as cold, if ε stays close to the thermal energy (3/2)kT in the fields under consideration. This is the case for gases with vibrational and rotational energy levels, the excitation of which causes inelastic energy losses to the drifting electrons. Cold gases are of interest since they exhibit the smallest possible diffusion. For gas mixtures with number densities ni (N =  ni ), the effective σ and  are given by σ = Σni σ1 /N, and Δσ = Σni Δi σ1 /N.

(4.17)

At low E, drift velocities rise with electric field. Some (e.g. CH4 and Ar Ar - CH4 mixtures) go through a maximum, decrease and may rise again. Drift velocities are shown in Fig. 4.6 for some gases [30, 31].

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H. J. Hilke and W. Riegler

Drift of Ions Ions of mass mi acquire the same amount of energy between two collisions as electrons but they lose a large fraction of it in the next collision and their random energy thus remains close to thermal energy. On the other hand the direction of their motion is largely maintained. The result is a much smaller diffusion compared to electrons and constant mobility up to high fields (to ~20 kV/cm atm for A+ ions in A). In the approximation for low E field, the random velocity is considered thermal, i.e. the relative velocity cre1 between the ion and the gas molecules of mass M, which determines τ , is   2 2 2 −1 (4.18) = cion + cgas = 3kT m−1 cre1 i +M An argumentation similar to the one followed for electrons [19] leads to  1/2 −1 u = m−1 + M (1/3kT )1/2 eE/ (Nσ ) i

(4.19)

The ion drift velocity at low fields is thus proportional to the electric field. Typical values at 1 atm are around u = 4 m/s for E = 200 V/cm, to be compared with a thermal velocity around 500 m/s. In the other extreme of very high fields, where thermal motion can be neglected, one finds the drift velocity being proportional to the square root of E. Measurements on noble gas ions [32] in their own gas clearly show both limits with a transition between them at about 15 − 50 kV/cm atm; see Fig. 4.7. As typical drift fields in drift chambers are a few hundred V/(cm atm), the ‘low field approximation’ is usually applicable, except in the amplification region. In a gas mixture it is expected that the component with the lowest ionization energy will rapidly become the drifting ion, independently of which atom was ionized in the first place. The charge transfer cross-section is in fact of similar magnitude as the other ion molecule scattering cross-sections. Even impurities rather low concentration might thus participate in the ion migration.

Magnetic Field Effects A simple macroscopic argumentation introduced by Langevin produces results which are a good approximation in many practical cases. The motion of a charged particle is described by mdu/dt = eE + e [u x B] − k u,

(4.20)

where m, e and u are the particle’s mass, charge and velocity vector, respectively; E and B are the electric and magnetic field vectors; k describes a frictional force proportional to −u.

4 Gaseous Detectors

103

Fig. 4.7 Drift velocities of singly charged ions of noble gases [32]

In the steady state du/dt = 0 and u/τ − (e/m) [u x B] = (e/m) E,

(4.21)

with τ = m/k. The solution for u is   *  ) + , u = (e/m) τ | E | 1/ 1 + ω2 τ 2 E∗ + ωτ E∗ x B∗ + ω2 τ 2 (E∗ B∗ ) B∗ , (4.22) where ω = (e/m)  B, and ω carries the sign of e and E∗ and B∗ are unit vectors. For ions, ωτ ≈ 10−10 Therefore, magnetic fields have negligible effect on ion drift. For electrons, u is along E, if B = 0, with u = (e/m) τ E.

(4.23)

This is the same relation as the one derived from the microscopic picture (4.11), which provides the interpretation of τ as the mean time between collisions. For large ωτ , u tends to be along B, but if EB = 0, large ωτ tums u in the direction of ExB.

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Two special cases are of practical interest for electron drift:

E orthogonal to B With EB = 0 and choosing E = (Ex , 0, 0) and B = (O, 0, Bz ), we get   ux = (e/m) τ | E | / 1 + ω2 τ 2 ,   uy = − (e/m) τ ωτ | E | / 1 + ω2 τ 2 , uz = 0,

(4.24)

tgψ = uy /ux = −ωτ.

(4.25)

and

The latter relation is used to determine ωτ , i.e. τ , from a measurement of the drift angle ψ, the so-called Lorentz angle. In detectors, this angle increases the spread of arrival times and sometimes also the spatial spread. A small ωτ would, therefore, be an advantage but good momentum resolution requires usually a strong B. The absolute value of u is  −1/2 | u |= (e/m) τ | E | 1 + ω2 τ 2 = (e/m) τ | E | cos ψ.

(4.26)

This means that, independent of the drift direction, the component of E along u determines the drift velocity (Tonks’ theorem). This is well verified experimentally.

E Nearly Parallel to B This is the case in the Time Projection Chamber (TPC). Assuming E along z and the components BX  and By  < <  Bz , one finds in first order     ux /uz = −ωτ By /Bz + ω2 τ 2 Bx /Bz / 1 + ω2 τ 2 , and     uy /uz = ωxBx /Bz + ω2 τ 2 By /Bz / 1 + ω2 τ 2 .

(4.27)

In a TPC this will produce a displacement after a drift length L of δ x = Lux /uz and δ y = Luy /uz From measurements with both field polarities and different fields, BX , By and τ can be determined. If Bx and By can be neglected with respect to Bz , uz remains unaffected by B.

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4.2.2.2 Diffusion Due to the random nature of the collisions, the individual drift velocity of an electron or ion deviates from the average. In the simplest case of isotropic deviations, a pointlike cloud starting its drift at t = 0 from the origin in the z direction will at time t assume a Gaussian density distribution   N = (4πDt)−3/2 exp −r 2 / (4Dt) ,

(4.28)

with r2 = x2 + y2 + (z − ut)2 , D being the diffusion coefficient. In any direction from the cloud centre, the mean squared deviation of the electrons is σI = (2Dt)1/2 = (2Dz/u)1/2 = D ∗ z1/2 .

(4.29)

with D∗ called diffusion constant. In terms of the microscopic picture, D is given by D = λ2 / (3τ ) = cλ/3 = c2 τ/3 = (2/3) (ε/m) τ,

(4.30)

with λ being the mean free path, λ = cτ , and ε the mean energy. With the mobility μ defined by μ = (e/m) τ,

(4.31)

the mean energy ε can be determined by a measurement of the ratio D/μ: ε = (3/2) (D/μ) e.

(4.32)

Instead of ε, the characteristic energy εk = (2/3)ε is often used. The diffusion width σ x of an initially point-like electron cloud having drifted a distance L is determined by the electron energy ε: σX2 = 2Dt = 2DL/ (μE) = (4/3) εL/(eE)

(4.33)

This relation is used for the determination of D and ε. For a good spatial resolution in drift chambers, a low electron energy and high electric fields are required. The lower limit for ε is the thermal energy εth = (3/2)kT. In this limit, the relationship known as Einstein or Nernst-Townsend formula follows: D/μ = kT /e.

(4.34)

The minimum diffusion width is thus σx,mm 2 = (kT /e) (2L/E) .

(4.35)

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Fig. 4.8 Longitudinal and transverse diffusion constants for low electric fields [33]. The dashdotted line denotes the thermal limit

As can be seen in Fig. 4.8, this minimum is approached for ‘cold gases’ like Ar/CO2 up to E~150 V/cm at 1 atm, for ‘hot gases’ like Ar/CH4 only for much lower fields.

Anisotropic Diffusion So far, we have assumed isotropic diffusion. In 1967 it was found experimentally [34], that the longitudinal diffusion DL along E can be different from the transversal diffusion DT Subsequently it has been established that this is usually the case. For ions this anisotropy occurs only at high E. As in a collision ions retain their direction to a large extent, the instantaneous velocity has a preferential direction along E. This causes diffusion to be larger longitudinally. However, this high field region is beyond the drift fields used in practical detectors. For electrons a semi-quantitative treatment [35], restricted to energy loss by elastic collisions, shows that DL /DT = (1 + γ ) / (1 + 2γ ) with γ = (ε0 /v0 ) (δv/δ) .

(4.36)

It follows that longitudinal and transversal diffusion will be different, if the collision rate v depends on the electron energy ε. Figures 4.8, 4.9, 4.10 show measured diffusion for a drift of 1 cm for some common gas mixtures [30, 33, 36]. Simulated diffusion curves are compiled in [33].

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Fig. 4.9 Transverse diffusion for 1 cm drift in Ar/CH4 mixtures; CH4 % is indicated [36].

Fig. 4.10 Transverse and longitudinal diffusion for 1 cm drift up to higher E fields [30]

A magnetic field B along z will cause electrons to move in circles in the x-y projections in between collisions. The random propagation is diminished and the transverse diffusion will be reduced:   (4.37) DT (ω) /DT (0) = 1/ 1 + ω2 τ 2 .

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This reduction is essential for most TPCs with their long drift distances. A more rigorous treatment of averages [19] shows that different ratios apply to low and high B: D(0)/D(B) = 1 + ω2 τ12 D(0)/D(B) = C + ω2 τ22

for low B, and for high B.

(4.38)

This behaviour was indeed verified [37], by measuring D(B) over a wide range of B. In an Ar/CH4 (91/9%) mixture the data could be fitted with τ 1 = 40 ps, τ 2 = 27 ps and C = 2.8. The high field behaviour is approached above about 3 kg, close to ωτ = 1. The longitudinal diffusion remains unchanged: DL (ω) = DL (0). The effects of E and B combine if both fields are present.

4.2.2.3 Electron Attachment In the presence of electronegative components or impurities in the gas mixture, the drifting electrons may be absorbed by the formation of negative ions. Halogenides (e.g. CF4 ) and oxygen have particularly strong electron affinities. Two-body and three-body attachment processes are distinguished [38]. In the two-body process, the molecule may or may not be broken up:   e− + AX → Ax − ∗ → A (or A∗ ) + X− or X− ∗ or e− + AX → Ax − ∗ → AX− + energy.

(4.39)

The attachment rate R is proportional to the density N: R = cσ N

(4.40)

for an electron velocity c and attachment cross-section σ The rate constants of freons and many other halogen-containing compounds are known [39]. The best known three-body process is the Bloch-Bradbury process [40]. In this process, an electron is attached to a molecule through the stabilizing action of another molecule. It is important for the attachment of electrons with energy below 1 eV to O2 , forming an excited unstable state with a lifetime τ of the order of 10−10 s. A stable ion will be formed only if the excitation energy is carried away during τ by another molecule. The attachment rate is proportional to the square of the gas pressure, as it depends on the product of the concentrations of oxygen and of the stabilizing molecules [19]: R = τ ce c2 σ1 σ2 N (O2 ) N (X) .

(4.41)

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Here ce is the electron velocity, c2 the relative thermal velocity between O2 and X In an Ar/CH4 (80/20%) mixture at 8.5 atm with an O2 contamination of 1 ppm, an absorption of 3%/m was measured at a drift speed of 6 cm/μs.

4.2.3 Avalanche Amplification 4.2.3.1 Operation Modes Gas detectors generally use gas amplification in the homogeneous field of a parallel plate geometry or, more frequently, in the inhomogeneous field around a thin wire. We shall start with the discussion of the second case. Near a wire with a charge qs per cm, the electric field at a distance r from its centre is E = qs / (2πε0 r) .

(4.42)

When raising the field beyond the ionization chamber regime, in which all primary charges are collected without any amplification, at some distance from the wire a field is reached, in which an electron can gain enough energy to ionize the gas and to start an avalanche. The avalanche will grow until all electrons have arrived on the anode wire. For a gas amplification A of 1000 ~ 210 , some 10 ionization generations are required. As the mean free path between collisions is of the order of microns, the field to start an avalanche has to be several times 104 V/cm. This is usually achieved by applying a voltage of a few kV to a thin wire, with a diameter in the 20 − 50 μm range. Besides ionization, excitation will always occur and with it photon emission. A fraction of these photons may be energetic enough to produce further ionization in the gas or on the cathode. Only those photons which ionize outside the radius rav of the moving electron avalanche may be harmful, as their avalanches will arrive later. If γ called the second Townsend coefficient, is the probability per ion pair in the first avalanche to produce one new electron, and if A denotes the amplification of the first avalanche, breakdown will occur for Aγ > 1.

(4.43)

In this case the first avalanche will be followed by a bigger one, this by an even bigger and so on, until the current is limited by external means. If Aγ < < 1, Aγ gives the probability for producing an after-discharge. If a photoionization takes place inside rav , the effect will be an increase of A. The resulting need to suppress far-traveling photons produced in the rare gases is the reason for the use of ‘quench gases’ like Methane, Ethane, CO2 , etc., which have large absorption coefficients for UV photons.

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The positive ions produced in the avalanche have too little energy to contribute to the ionization in the avalanche. They will move slowly to the cathode(s), where they get neutralized but where rare gas ions may also liberate additional electrons. The addition of the quencher reduces this risk significantly, as its recombination energy can be dissipated in other ways, e.g. by disintegration. This explains why more complex molecules provide higher protection. Up to a certain value Ap , one has a proportional regime: the signal produced will on average be proportional to the number of primary electrons. The amplification will rise approximately exponentially with voltage. The azimuthal extension of the avalanche around the wire will grow with amplification and eventually the avalanche will surround the wire. When the field is raised above this proportional regime, space charge effects will set in. The space charge of the positive ions—moving only very slowly compared to the electrons—will reduce the field at the head of the avalanche and the amplification will rise more slowly with voltage and will no longer be proportional to the primary ionization. In addition, space charge effects will depend on the track angle with respect to the wire and on the density of the primary ionization. This is the so-called limited proportionality regime. Increasing the field further, the positive space charge may produce additional effects. Near the avalanche tail the electric field is increased. If the absorption of UV photons in the quench gas is high, the photons may ionize this high field region and start a limited streamer moving backwards by starting avalanches further and further away from the sense wire. As the electric field at large radius weakens, this development will stop after typically 1–3 mm. The total charge is almost independent of the primary charge starting the streamer. The process depends quite strongly on experimental conditions. An example is presented in Fig. 4.11, which shows a steep step from the proportional regime [41]. In the narrow transition zone one finds a rapid change of the ratio of streamer/proportional signal rates. In other experimental conditions a smoother transition has been observed. If the absorption of the UV photons is weak, photons travel further and avalanches may be started over the full length of the wire, leading to the Geiger mode, if the discharge is limited by external means.

4.2.3.2 Gas Gain With multiplication, the number n of electrons will grow on a path ds by dn = n α ds,

(4.44)

where α is the first Townsend coefficient. Ionization growth is obviously proportional to the gas density p and depends on the ionization cross-sections, which are a function of the instantaneous energy ε of the electrons. This energy itself is a function of E/p. The relationship between α and E is, therefore, given in the form α/p as function of E/p or for a specific temperature as α/p(E/p). Figure 4.12 gives some

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Fig. 4.11 Pulse-height transition from limited proportionality to limited streamer mode [41]

examples of measurements [42]; it shows the strong increase of α with electric field in the region of interest to gas detectors, up to about 250 kV/cm. No simple relation exists for α as function of electric field E, but Monte Carlo simulation has been used to evaluate α. Figure 4.13 shows an example [27]. For the lower field values there is reasonable agreement with measurement. The discrepancy at the highest fields is attributed to photo-and Penning-ionization not being included. The amplification A in the detector is obtained by integration  A = n/n0 = exp

 α(s) ds = exp

α(E)/ (dE/ds) dE,

(4.45)

from Emin , the minimum field to start the avalanche, to the field E(a) on the wire. Emin e is equal to the ionization energy of the gas molecules divided by the mean free path between collisions. Near the wire and far from other electrodes, the field is E(r) = qs / (2πrε0) ,

(4.46)

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Fig. 4.12 Examples of measured ‘First Townsend coefficient’ α in rare gases [42]

Fig. 4.13 Simulated ‘First Townsend coefficient’ α in Ar/CH4 mixtures at 1 atm (100–0 means 100%Ar). Measured values are indicated as circles [27]

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where qs is the charge per cm. Therefore,  A = exp

  qs α(E) dE/ 2πε0 E 2 .

(4.47)

Two approximations in particular have been used to describe practical cases. The early Korff model [43] uses the parameterization α/p = A exp (−Bp/E) ,

(4.48)

with empirical constants A and B depending on the gas. In the Diethorn approximation [44], α is assumed to be proportional to E. One then obtains for a proportional tube with wire radius a and tube radius b  ln A = (ln 2/ ln (b/a)) (V /ΔV ) ln V / (ln (b/a) aE min ) ,

(4.49)

where the two parameters Emm and V are obtained from measurements ofA at various voltages and gas pressures. Emm is the minimum E field to start the avalanche and eV the average energy required to produce one more electron. Emm is defined for a density p0 at STP. For another density Emm (p) = Emm (p0)(p/p0 ). A list for Emm and V for various gases is given, e.g., in [19]. Reasonable agreement with the experimental data is obtained; discrepancies show up at high A.

4.2.3.3 Dependence of Amplification on Various Factors The gas amplification depends on many operational and geometrical parameters. Some examples are:

Gas Density The Diethom approximation gives dA/A = − (ln 2/ ln (b/a)) (V /Δ∇) (dp/p) →= (5-8) dp/p

(4.50)

typically.

Geometrical Imperfections The effects will obviously depend on the geometry and the operation details. An early publication [45] gives analytic estimates of the effects of wire displacements and variations in wire diameter. In a typical geometry dA/A~2.5 dr/r, where r is the wire radius; dA/A~9gap/gap.

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Edge Effects Near edges, the electric field is reduced over distances similar to the gap between the electrode planes. It can be recovered largely by additional field shaping lines on the edges [46].

Space Charge Due to the low velocity of the positive ions (falling off as 1/r from >1mm/μs at r = a), space charge will build up at high particle fluxes and lower the avalanche amplification. In drift tubes the voltage drop due to the space charge from a given particle flux is proportional to the third power of the tube radius. A smaller radius thus improves the rate capability drastically.

4.2.3.4 Statistical Fluctuations of the Amplification In the proportional regime, the amplification A is simply defined by A = n/nT and one assumes that each of the nT initial electrons produces on average the same A ion pairs. We define P(n) as the probability to produce n electrons in the individual avalanche with mean A and variance σ 2 If nT > > 1 and if all avalanches develop independently, it follows from the central limit theorem that the distribution function F(n) for the sum of the nT avalanches approaches a Gaussian with mean n = nT A and variance S2 = nT σ 2 , independent of the actual P(n). On the other hand, for detection of single or a few electrons, knowledge of the individual P(n) is required. For a parallel plate geometry calculations [47] agree well with measurements [48]. The distributions found theoretically [49] and experimentally [50] for the strong inhomogeneous field around a thin wire also look similar and approach Polya distributions (Fig. 4.14). For these distributions (σA / < A >)2 = f, with f ≤ 1.

(4.51)

The limiting case f = 1 is an exponential distribution (Yule-Furry law) P (A) = (1/ < A >) exp (−A/ < A >) .

(4.52)

Experimental results point to f = 0.6 − 1.0. Measurements with laser tracks [19], indicate that the r.m.s. width σ A of a single-electron avalanche is close to the mean, as it is for the Polya distribution with f = l. An approximately exponential distribution for single-electron avalanches is also reported in [28, 48].

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Fig. 4.14 Polya distributions [22]

4.2.4 Signal Formation In wire chambers, signal formation is very similar to the one in the simplest geometry of a cylindrical tube with a coaxial wire, because most of the useful signal is produced in the immediate vicinity of the sense wire and the electric field around the sense wires in a MWPC can be considered as radial up to a radius equal to about one tenth of the distance between sense wires [45]. Signals are always produced by induction from the moving charges. Ramo [51] and Shockley [52] have shown that in general the current IR induced on the readout electrode R is given by IR = −q Ew v,

(4.53)

where q is the signed charge moving with the vectorial velocity v, and Ew is a vectorial weighting field, a conceptual field defined by applying + 1V on R and 0 V on all other electrodes. The unit of Ew is 1/cm. The actual v is calculated by applying the normal operation voltages, including possibly a B field. In the special case of a two electrode system like the wire tube, Ew = Eop /V, where Eop is the actual operating field obtained with the voltage V on R (the anode wire) and zero V on the cathode.

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For the proportional tube with wire radius a and cathode radius b, Eop and Ew are obviously radial with Eop = V / [r ln (b/a)] .

(4.54)

We assume constant mobility μ for the positive ions. Therefore v + (t) = μV / [r(t) ln (b/a)] .

(4.55)

For an ion starting at t = 0 from r = r1 , r(t) = r1 (1 + (t/t0 ))1/2 with t0 = r12 ln (b/a) / (2μV ) .

(4.56)

The maximum time for an ion to drift from a to b is T + max = (b/a)2 t0 , as (b/a)2 >> 1.

(4.57)

The induced current I+ is I + = −q Ew v+ < 0,

(4.58)

as v+ is parallel to Ew . For the integrated charge Q, one obtains +

Q (t) =



 I dt =



I 1/v

+



 dr =

−qE w dr.

(4.59)

Integration from r1 to r2 gives Q+ 1→2 = −q ln (r2 /r1 ) / ln (b/a) , with q > 0 and r2 > r1 ,

(4.60)

For an electron one obtains Q− 1→2 = +q | ln (r2 /r1 ) | / ln (b/a) , with q < 0 and r2 < r1 ,

(4.61)

as v is antiparallel to Ew . We shall give numbers for a typical proportional tube with a = 10 μm, b = 2.5 mm, Eop (r = a) = 200 kV/cm, μ+ = 1.9 atm cm2 /(Vs), v− ≈ 5 · 106 cm/s and—to estimate the gas amplification A—the Diethom parametrization α = (ln2/∇)E and Emin = V/(rmm ln (b/a)), taking for an Ar/CH4 (90/10) mixture V = 23.6 V and Emin = 48 kV/cm [19], p. 136. Here rmm is the starting radius for the avalanche and + = Emin the minimum field permitting multiplication. We obtain: t0 = 1.3 ns, Tmax 82 μs, rmin = 42 μm, A = 4400. The last electron will be collected in a very short time of about 0.6 ns, the vast majority even faster. Half of the electrons move only about 2 μm, the next 25%

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some 4 μm and so on. A rough estimate of the induced electron charge signal is, therefore, Q− total = q ln (14/10) / ln (2500/10) = 0.06 q.

(4.62)

Only about 6% of the total induced signal is due to the movement of the electrons, the rest from the ions, if one integrates over the full ion collection time. In practice, however, one mostly uses much faster integration. The long tail in the signal caused by the very slow ion movement has to be corrected for by electronic pulse shaping to avoid pile-up at high rates (see Sect. 4.69). If one uses fast pulse shaping, say 20 ns integration, only a fraction of the ion charge will be seen: an ion starting at r1 = a, reaches r2 = 40 μm in 20 ns and induces about 25% of its charge. That means: with 20 ns pulse shaping, one may expect to see an effective charge of about 30% of the total charge produced, of which one fifth is due to the electrons.

4.2.5 Limits to Space Resolution The space resolution σ X obtained from a single measurement of the anode wire signals in a multi-wire proportional chamber is given by the separation s of the √ wires: σ x = s/ 12. The minimum practical s for small chambers is 1 mm. The best resolution is thus about 300 μm. Significantly better resolution may be obtained either from ‘centre of gravity’ determination or from the electron drift times in drift chambers.

4.2.5.1 ‘Centre of Gravity’ Method In this method one uses the signals induced on cathode strips or pads, see Fig. 4.19. The rms width of the induced charge distribution is comparable to the anode-cathode gap d. If one chooses a strip width of (1–2)d, one obtains signals above threshold on typically 3–5 strips. Depending on the signal to noise ratio, a resolution of typically (1–5)% of the strip width is achieved, i.e. about 40 − 100 μm. This method is used for the read out of TPCs, as well as for high precision cathode strip chambers, see e.g. [15, 16].

4.2.5.2 Drift Time Measurement The main contributions to the error of the drift time determination come from electronics noise, electron clustering, δ-rays, diffusion. This assumes that additional effects on the space-time correlation including magnetic field corrections, gain variations, gas contamination and others are kept small by careful construction and calibration. Figures 4.16 and 4.24 (right) show typical results. Electronic noise

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contributes a constant error. Near the anode wire, the effects of the clustering of the primary charges adds a significant error. At large distances from the anode, the contribution from diffusion grows as square root of distance. Resolutions achieved are typically 50 − 200 μm. A detailed discussion of limits to space resolution is presented in [19] and for the particular case of proportional tubes in [15].

4.2.6 Ageing of Wire Chambers Deterioration of performance with time has been observed since the early days of gas detectors but has gained importance with the ever increasing radiation loads due to the demand for higher detection rates over long periods. Typical effects of ageing are: pulse height decrease, a broadening of the energy resolution and increase in dark current, in the extreme also electrical breakdown or broken wires. An enormous number of studies has been carried out. They are well documented in the proceedings of workshops [54] and several reviews [55]. Upon opening of damaged chambers, deposits have been observed on anode wires and/or on cathodes. On the wire they can take any form from smooth layers to long thin whiskers [56], see Fig. 4.15. On the cathodes, deposits usually consist in spots of thin insulator. Defects of this latter kind can often be correlated with a discharge pattern, which may be interpreted as Malter effect [57]: under irradiation, charges build up on the insulator until the electric field is strong enough to extract electrons from the cathode through the layer into the gas where they initiate new avalanches. The facts that the buildup time decreases with higher ionization rate and that the discharges take some time to decay after irradiation is timed off, give support to this explanation, as does the observation that addition of water vapour is reducing the discharges, probably introducing some conductivity. Analysis of the layers and whiskers on the anode wires often indicate carbon compounds, more surprisingly also often silicon, sometimes other elements: Cl, O, S. The aging results are often characterized by a drop in pulse height PH as function of integrated charge deposition in Coulomb per cm wire, although it was found in some cases that the rate of the charge deposition has an influence. Typical values with classical gas mixtures containing hydrocarbons are ΔP H /P H ∼ 0.01 − 0.1%/0/mC/cm for small detectors, ΔP H /P H ≈ 0.1 − 1%/mC/cm

for large detectors.

(4.63)

It is obvious that the control of ageing is one of the major challenges for the LHC experiments, possibly even the major one. Unfortunately, however, it has not been possible to establish a common fundamental theory, which could predict lifetimes of a new system. On the other hand, the

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Fig. 4.15 Examples of deposits on 20 μm anode wires after strong irradiation [56]

reasons for ageing in particular circumstances have been elucidated and the studies permit to establish some general rules on how for improving the chances for a longer lifetime: • Many materials have to be avoided, in the gas system, in the detector and during the construction: Si compounds, e.g. in bubbler oils, adhesives, vacuum grease or protection foils, PVC tubing, soft plastics in general, certain glues and many more. The workshop proceedings and reviews mentioned present details, also on materials found acceptable. • For the highest radiation loads, up to a few C/cm, gas components most frequently used in the past, namely hydrocarbons like Methane, Ethane, etc., should be avoided. Indeed, the LHC experiments make use of them only exceptionally. There remains only a very restricted choice of acceptable gases: mixtures of rare gases and CO2 and possibly N2 , CF4 or DME. CF4 is offering high electron drift

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velocities and has proven to be capable under certain conditions to avoid or even to etch away deposits, in particular in the presence of minute Si impurities. But its aggressive radicals may also etch away chamber components, especially glass [15]. In any case the water content has to be carefully controlled to stay below 0.1%, if CF4 is used, to avoid etching even of gold-plated wires. Also DME, offering low diffusion, has in some cases provided long lifetime. It has, however, shown to attack Kapton and to be very sensitive to traces of halogen pollutants at the ppb level. • During production, high cleanliness has to be observed, e.g. to avoid resistive spots on the cathodes. The sense wire has to be continuously checked during wiring to assure the required quality of its geometrical tolerances and of the gold plating. • The gas amplification should be kept as low as possible. • In any case, a final detector module with the final gas system components should be extensively tested under irradiation. As an accelerated test is usually required for practical reasons, to obtain the full integrated charge for some 10 years of operation in a 1 year test, an uncertainty un- fortunately will remain, because a rate dependence of the ageing cannot be excluded.

4.3 Detector Designs and Performance 4.3.1 Single Wire Proportional Tubes Despite the revolution started with the multiwire proportional chambers (MWPC), single wire tubes are still widely used, mostly as drift tubes. They have circular or quadratic cross-section and offer independence of the cells, important, e.g., in case wire rupture. We present three examples. ATLAS has chosen for the muon system large diameter (3 cm) aluminum tubes operated with Ar/CO2 (93/7%) at 3 atm, with the addition of about 300 ppm of water to improve HV stability. A pair of 3 or 4 staggered tube layers, separated by a support frame, form a module. The disadvantages of the gas mixture, a nonlinear space-drift time relation and relatively long maximum drift time, had to be accepted in order to obtain a high radiation tolerance. The spatial resolution for a single tube under strong γ - irradiation producing space charge is shown in Fig. 4.16. An average resolution per tube of 80 μm is expected with a maximum background rate of 150 hits/cm2 s. With a relative positioning of the wires during construction to 20 μm, an adjustment of the tube curvature to the gravitational wire sag and a relative alignment and continuous monitoring of the pair of layers inside a chamber, a combined resolution for the 6–8 1ayers of ~35 μm is aimed at. These chambers provide only one coordinate, the other one being measured in other subdetectors of the experiment.

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Fig. 4.16 ATLAS MDT drift tubes: space resolution as function of impact radius and background rates. Expected rates are ≤150 hits/cm2 s [15]

Fig. 4.17 LHCb straw tubes: (a) Winding scheme. (b) Details of the double foil. Kapton XC is on the inside [17]

A second type of tube design, straw tubes, has become very popular since a number of years. Straw tubes offer high rate capability due to small diameters and relatively little material in the particle path. In LHCb, where local rates (near one end of the wire) up to 100 kHz/cm have to be handled, an internal diameter of 4.9 mm was chosen [17], with a construction shown in Fig. 4.17. Two strips of thin foils are wound together with overlap. The inner foil, acting as cathode, is made

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of 40 μm carbon doped polyimide (Kapton-XC), the outer is a laminate of 25 μm polyimide, to enhance the gas tightness, and 12.5 μm aluminum, to ensure fast signal transmission and good shielding. The tubes are up to 2.5 m long and have the 25 μm wire supported every 80 cm. Staggered double layers tubes are glued to light support panels to form modules up to 5 m long. An average spatial resolution of a double layer below 200 μm was measured with Ar/CO2 (70/30%). In a station, 4 double layers are aligned along ◦ 0, + 5, − 5, 0 , thus providing a crude second coordinate measurement. In another design for ATLAS [15], mechanical strengthening of 4 mm straws is achieved with carbon fibers wound around the tubes and straightness by supporting them every 25 cm with alignment planes vertical to the straws. This construction reduces the material along the radial tracks, which is essential for the role of the straws to detect transition radiation originating in fibers stacked in between the tube layers. This role also determines the need for Xe in a mixture of Xe/CO2 /O2 (70/27/3%), the oxygen addition increasing the safety margin against breakdown. Another quasi-single wire design is that of plastic streamer tubes usually called Iarocci tubes. Because they are easy to construct in large size and cheap, they have been widely used, especially as readout planes in hadron calorimeters. A plastic extrusion with an open profile with typically 8 cells of lxl cm2 and a PVC top plate, is coated on the inside with graphite with a minimum resistivity of 200 k /square. All this is slid into a plastic box, which serves also as gas container. For stability, wires are held by plastic spacers every 50 cm. Using a thick wire of 100 μm, self-quenching streamers are initiated in a gas containing a strong quencher, typically isobutane in addition to Ar. Electrodes of any shape placed on one or both external surfaces pick up the rather strong signals. The dead time is long but only locally, so that particle rates up to 100 Hz/cm2 can be handled.

4.3.2 Multiwire Proportional Chambers (MWPC) Already 1 year after the invention of the multiwire proportional counter (MWPC) by Charpak in 1968, a system of small chambers was used in an experiment [58], another year later a large chamber 2 m × 0.5 m had been tested with Ar/Isobutane [59]. A number of developments like bi-dimensional readout were discussed [60]. In 1973 already, a large system of MWPC containing 50,000 sense wires had been constructed for an spectrometer at the ISR, the Split-Field Magnet (SFM) [61]. All these chambers had a geometry for the sense wires and gap size similar to the original design, shown in Fig. 4.18. The SFM chambers of 2 × 1 m2 contained three light support panels forming two amplification gaps of 2 × 8 mm each, one with vertical, the other with horizontal wires of 20 μm diameter. The cathodes on the panels were sprayed with silver paint ◦ to provide readout strips 5.5 cm wide and at angles of ±30 , to resolve ambiguities. Special emphasis was put on high precision with a light construction, including frames of only 5 mm thickness. A total thickness of 1.7% of a radiation length

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Fig. 4.18 Design of the first multiwire proportional chamber [8]

Fig. 4.19 Principle of ‘centre of gravity’ cathode measurement of cathode strip signals

per chamber was achieved. The stringent quality demands can be inferred from the definition of the efficiency plateau: the ‘beginning of plateau’ was defined as efficiency ε 99.98% and the end by 10 times the ‘normal noise’, corresponding to cosmics rate With this tight definition, the measured plateau length for a chamber was 50 − 100 V with the magic gas mixture of Ar/isobutane/freon/methylal (67.6/25/0.4/7%). In the following decades, drift chambers imposed themselves more and more, but MWPCs remained valid options, especially when speed was more important than high spatial resolution. Thus three of the four LHC experiments employ MWPCs, for triggering and momentum measurements. All of them make use of the signals induced on the cathode strips. In ATLAS and CMS, who call their chambers cathode strip chambers (CSC), a high precision measurement is obtained in the bending coordinate by determining the ‘centre of gravity’ of the strip signals, see Fig. 4.19. In ATLAS, each third strip is read out (pitch ~5.5 mm), leaving two strips floating,

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and a resolution of 60 μm is obtained [15, p. 178]; in CMS, 75 μm resolution is achieved with each strip read out at a minimum pitch of 8.4 mm [16, p. 197]. In LHCb, spatial resolution is secondary to fast timing and high efficiency for a five-fold coincidence trigger. Adjustment to the requirements on spatial resolution, which change strongly with radius, is achieved by forming readout pads of variable size (0.5 × 2.5 − 16 × 20 cm2 ) on the cathodes and by grouping sense wires [17, p. 130].

4.3.3 Drift Chambers Already in the very first publications, the basic two types of drift chambers were described: (i) with the drift volume, through which the particles pass, separated from the amplification volume [9] and (ii) a geometry, in which the particles pass directly through the volume containing the anode wires alternating with field wires to improve the drift field [10], see Fig. 4.20. The first design finally evolved into the TPC, the second into a large number of different designs. One can differentiate between planar and cylindrical geometries.

4.3.3.1 Planar Geometries Most planar geometries are rather similar to each other. To obtain a more homogeneous drift field, additional field shaping electrodes are introduced, see Fig. 4.21. Also shown is a recent example, one element of a layer for the Barrel Muon system of CMS. The space resolution per layer is about 250 μm. One muon station consists of 2 × 4 layers of such tubes fixed to an aluminum honeycomb plate. The other ◦ coordinate is provided by a third set of 4 layers oriented at 90 . The central detector of UAl used a special arrangement, see Fig. 4.22. In a horizontal magnetic field, at right angle to the beam, a cylinder 6 m long and 2.2 m in diameter is filled with planar subelements. In the central part, vertical anode planes

Fig. 4.20 First two drift chamber designs. Left: separate drift and amplification gaps [9]. Right: Common drift and amplification volume. The additional field wires improve the drift field [10]

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Fig. 4.21 Planar arrangement with field shaping electrodes. Right: cross-section of large drift tubes for CMS [16]

◦

Fig. 4.22 Horizontal view of a reconstructed event in the central detector of UAl. The first Z decay observed [19]

ultimate with cathode planes, leaving 18 cm drift spaces. These planes are horizontal in the two ends. Charge division is used for the coordinate along the wire. The average point accuracy along the drift direction was 350 μm.

4.3.3.2 Cylindrical Geometries Many different arrangements have been worked out. Figure 4.23 shows an example of a wire arrangement and electron drift lines following the electric field in the absence of a magnetic field. The change of the electron drift in a magnetic field in a similar cell is indicated to the right. Figure 4.24 (left) shows the conceptual design of the drift chamber for OPAL [53], a wire arrangement called Jet Chamber. The left-right ambiguity is solved by staggering the sense wires alliteratively by ±100 μm. The measured space resolution in rφ for a single wire is presented in Fig. 4.24 (right). The figure shows the typical dependence on the distance r from the sense wire: for small r, the primary ion statistics dominates, at large r diffusion. In addition, there is a constant contribution from the noise of the electronics. The coordinate along the wires is

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Fig. 4.23 Two multi-layer wire arrangements and electron drift lines without (left) and with (right) magnetic field

Fig. 4.24 Jet Chamber. Left: conceptual design with staggered sense wires. Right: Space resolution obtained in OPAL with 4 atm [53]

obtained from charge division by using resistive sense wires and read-out on both ends of the wires: a resolution of about 1% of the wire length is reached. In other designs the second coordinate is obtained from orienting successive layers in stereo angles. Sometimes relative timing with read-out of both ends of the wire is used, providing again a resolution of about 1% of the wire length.

4.3.3.3 Time Projection Chambers (TPC) The TPC concept proposed by Nygren [62] in 1974 for the PEP4 experiment [63] offered powerful pattern recognition with many unambiguous 3-D points along a track and particle identification by combining dE/dx information from many samples with momentum measurement. Originally proposed to resolve jets at a low

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Fig. 4.25 Conceptual design of the STAR TPC operating at RIC [64]

energy e+ e− collider, the TPC design has proven years later to be the most powerful tracker to study central heavy ion collisions with up to several thousand particles in an event, at more than 100 events per second. The basic design elements have hardly changed over the years. Cylindrical field cages provide a homogeneous electric field between the central electrode and the planar wire chambers at both ends; see Fig. 4.25 for the conceptual design of the latest TPC in operation, the STAR TPC at RHIC [64]. The typical gas mixture is Ar/CH4 , which offers high drift velocity at low electric field and low electron attachment. The electrons from the track ionization drift to one of the two endcaps. They traverse a gating grid and a cathode grid before being amplified on 20 μm anode wires, separated with field wires. The avalanche position along the wires is obtained from measuring the centre of gravity of pulse heights from pads of the segmented cathode beneath. Figure 4.26 shows the electric field lines for a closed and an open gating grid. Gating is essential for the TPCs with their long drift length, to reduce space charge build-up. The gate is only opened on a trigger. All TPCs except PEP4 and TOPAZ operated at latm and profit from a strong reduction of lateral diffusion due to the factor ωτ ~5 in the strong magnetic field B oriented parallel to the electric field E. Higher pressure is rather neutral: ωτ decreases, but more primary electrons reduce relative fluctuations and thus ExB and track angle effects. Typical point resolutions in rΦ range from 150 to 200 μm at the e+ e− colliders [65]. Figure 4.27 shows a reconstructed Pb–Pb interaction observed in STAR. All TPCs except STAR and ALICE use the signals from the anode wires for dE/dx information. In STAR and ALICE, all information is taken from the pads, some 560,000 in ALICE [14]. Pressure improves dE/dx and the PEP4 TPC operating at 8.5 atm produced the best dE/dx resolution despite a smaller radius [65].

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Fig. 4.26 TPC wire chamber: electric field lines for a closed (a) and open x [cm] gating grid (b)

Fig. 4.27 A reconstructed Pb–Pb interaction observed in the STAR TPC [64]

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4.3.4 Parallel Plate Geometries, Resistive Plate Chambers (RPCs) Parallel plate devices offer fast response, as there is no drift delay and the avalanche amplification starts immediately. Keuffel’s spark counter [6] featured two metal electrodes at millimeter distance in a gaseous atmosphere, where the primary electrons deposited in the gap provoke a fast discharge and therefore a detectable signal. By the end of the 1960s the spark counters had arrived at time resolutions around 100 ps, the rates however were limited to 1 kHz and small areas of about 30 cm2 , since after each discharge the entire counter was insensitive during the recharge time of typically a few hundred microseconds. Parallel Plate Chambers (PPCs) use the same geometry but operate below the discharge voltage. The avalanche therefore induces a signal but does not create a discharge, allowing a rate capability of 100 kHz for a 80 cm2 detector [6]. Still, the fact that the detector mechanics and especially the detector boundaries have very carefully controlled to ensure stability, limits this detector to a rather small surface. The Pestov spark counter [66] uses the same parallel plate geometry, with one electrode made from resistive material having a volume resistivity of ρ = 109 − 1010

cm. The charge deposited locally on this resistive layer takes a time of τ ≈ ρε to be removed, where ε is the permittivity of the resistive plate. This time is very long compared to the timescale of the avalanche process, the electric field is therefore reduced at the location of the avalanche, avoiding a discharge of the entire counter. This allows stable operation of the detector at very high fields and particle rates. A counter with a size of 600 cm2 and a gas gap of 1 mm, operated at atmospheric pressure achieved a time resolution of CD , i.e. a low input impedance, is important because when Ceff is only of the same order of magnitude as the detector capacitance CD the charge is incompletely transferred to the electronics. This results in a loss of sensitivity and possibly crosstalk within the detector to neighbouring channels. Turning now to the question of measurement precision, respectively noise in the detector-amplifier system, we remark that it is customary to represent the effect of all amplifier noise sources by a single noise voltage Un placed at the input (Fig. 5.15). As this noise voltage generator is in series with detector and amplifier it is called serial noise. The presence of the serial noise voltage Un will result in an output voltage even if there is no signal charge present. For an evaluation of the serial noise charge, it is easiest to consider the charge necessary to compensate for the effect of the noise voltage, such that the output voltage remains at zero. The value can be immediately read from Fig. 5.15: Qn = Un (CD + Cin + Cf ) = CT Un with CT the total “cold” input capacitance. Notice that the serial noise is generated in the amplifier, the influence of the detector is due to the capacitive load at the amplifier input only. The detector itself produces noise due to statistical fluctuations of its leakage current I. This parallel

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Fig. 5.15 The effect of amplifier serial noise in a detector-amplifier system

noise is represented by a noise current source of density d/df = 2I·q in parallel to the detector capacitance CD . To estimate the charge measuring precision one has to follow separately signal and noise through the complete readout chain and compare their respective output signals. The signal produced by the amplifier will usually not be used directly; it will be further amplified and shaped, in order to optimize the ratio of signal to noise and to reduce the interference between subsequent signals. We will only consider a few very simple cases, the simplest being an idealized charge-sensitive amplifier followed by an RCCR filter. For a more elaborate treatment, the reader is referred to the literature (e.g. [15]). The arrangement of a CSA followed by an RCCR filter is shown in Fig. 5.16. The output of the CSA is a voltage step given for very high amplification as Q/Cf . The shaper does an RC integration followed by a CR differentiation. This procedure results in a signal peak, which for the same integration and differentiation time constant τ = R1 C1 = R2 C2 has the shape Uout (t) = (Q/Cf )·(t/τ )·exp(−t/τ ) with a peak value Upeak = (Q/Cf )·exp(−1). The height of this peak is a measure of the signal charge. Superimposed on the signal is the noise voltage, and we are interested in the signal-to-noise ratio, which is defined as the ratio of the height of the peak value to the root-mean-square value of the noise voltage measured at the same point in the circuit. In order to find the noise voltage at the output, each noise source in the circuit has to be traced to the output and the resulting voltages added in quadrature. Doing so, one finds the important result that, for white (thermal) serial noise, the ratio of noise to signal (N/S) decreases with the square root of the shaping time constant τ , while for 1/f noise this ratio remains constant. Parallel noise, given as a time integral over current fluctuations, increases with the square root of the shaping time. More sophisticated continuous time filtering methods use (for example) Gaussian shape filtering, which can be approximated by several sequential RC integration and differentiation steps. Especially important in integrated electronics are the

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Fig. 5.16 Noise filtering and signal shaping in an RCCR filter following a charge-sensitive amplifier (top). The two unity gain amplifiers have been introduced in order to completely decouple the functions of the CSA, the integration (RC) and the differentiation (CR) stages. The signal form is indicated for each stage (bottom)

techniques in which the output signal is sampled several times and mathematical manipulations of the samples are performed. This can be done either after the measurement by numerical processing or directly by the local readout electronics. In the latter case, it is usually achieved by using switched capacitor techniques for analogue algebraic manipulations. Common to both methods, however, is the need to sample the signal at fixed (or, at least, known) times with respect to its generation. Alternatively with frequent enough sampling, the arrival time of the signal can be extracted from the data and filtering can be done afterwards by selecting the relevant samples before and after arrival of the signal. In all cases, however, the fact that the three noise components (white serial, 1/f and white parallel noise) scale with the available readout time in the described way remains valid. As a further example we discuss double correlating sampling realized in switched capacitor technology which is most naturally realizable in integrated circuit technology. It is applicable if the signal arrival time is known in advance, as is the case for example in collider physics experiments. The circuit (Fig. 5.17) consists of two sequential charge-sensitive amplifiers connected by a coupling capacitor Cs and switch Sc . Initially all switches are closed. Thus both CSAs have reset their input and output voltages to proper working conditions and a possible offset voltage between CSA1 and CSA2 is stored on capacitor Cs . The following operations are performed in sequence: (1) opening switch S1 at time t1 , resulting in an unwanted charge injection into the input of CSA1 and therefore an output voltage change that will be stored on capacitance Cs and thus made invisible to the input of CSA2; (2) opening of reset switch S2 . Any voltage change on the output of CSA1 (e.g. signal or noise) is also seen in the output of CSA2, amplified by the ratio Cs /Cf2 ; (3) signal charge Qs generation at time t3

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Fig. 5.17 Double-correlated sampling of the output of a charge-sensitive amplifier (CSA1) with the help of a coupling capacitor Cs and a second CSA2

changes the output of CSA1 by U1 = Qs /Cf1 and the output voltage by Uout = Qs (Cs /(Cf1 Cf2 )); and (4) opening of switch Sc at time t4 inhibits further change of the output voltage. The difference of the output voltage of CSA1 (amplified by Cf2 /Cs ) between times t2 and t4 remains present at the output of the circuit. Double correlated sampling suppresses the reset noise due to operating switch S1 and also suppresses low frequency noise but enhances the noise at higher frequencies. As a result white noise is not suppressed. This has to be done by limiting the frequency range of the amplifier. More sophisticated schemes of switched capacitor filtering, taking several samples (sometimes with different weights), have also been implemented.

5.7.3 Integrated Circuits for Strip Detectors The development of integrated detector readout electronics was initiated by the simultaneous requirements of high density, low power and low noise for use with silicon strip detectors in the tight space environment of elementary particle physics

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Fig. 5.18 Single channel readout schematics of the CAMEX64 strip-detector readout circuit

collider experiments. A variety of circuits has been developed for this purpose, the basic principle of essentially all of them being: (1) parallel amplification using a charge-sensitive amplifier at each input; (2) parallel signal filtering combined with second-stage amplification and parallel storage within capacitive hold circuits; and (3) serial readout through one single output channel. We will present only one of the developments [16]. This was not only one of the first to be started but is still in use and has been further developed for many important applications. The basic functional principle of a single channel is shown in Fig. 5.18. It consists of two charge-sensitive amplifiers, each followed by a source follower, and four sets of capacitors and switches that connect the output of the first amplifier with the input of the second amplifier. The circuit is rather similar to the one shown in Fig. 5.17, but the essential difference is the fourfold multiplication of the capacitive coupling between the amplifiers. In this way it is possible to perform fourfold double-correlated sampling at times that are shifted relative to each other. This procedure provides a good approximation to trapezoidal shaping, which means averaging the output over time intervals before and after signal arrival and taking the difference between the averaged samples. The switching sequence that performs this function is the following: (1) Close R1 and R2 . The charges on the feedback capacitances Cf1 and Cf2 are cleared. (2) Open R1 : some (unwanted) charge will be injected into the input by the switching procedure, producing an offset in U1. (3) Close and open in sequence S1 to S4 . The U1 offset values at the four times t1 to t4 will be stored on the four capacitors Cs . (4) Open switch R2 . A small offset voltage appears at the output. (5) Deposit signal charge Qsig at input. U1 changes by an amount of U1 = Qsig /Cf1 . (6) Close and open S1 to S4 in sequence at times t1 to t4 . A charge Cs U1 is inserted into the second amplifier at each sample. The total output voltage is 4Cs U1 /Cf2 = Qsig 4Cs /(Cf1 Cf2 ). The complete chip, containing 64 channels, also comprises additional electronics, as shown in Fig. 5.19. Three test inputs allow injection of a defined charge through test capacitors. Digital steering signals are regenerated by comparators. The

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Fig. 5.19 Block diagram of the CAMEX64 strip-detector readout chip

Fig. 5.20 Circuit diagram of the amplifier, including source follower and biasing circuit, of the CAMEX64 strip-detector readout chip

decoder switches one signal at a time on the single output line where a driving circuit for the external load is attached. A circuit diagram valid for all charge sensitive amplifiers used is shown in Fig. 5.20. The current in all transistors can be scaled by a reference bias current (Bias). The input (In) can be shorted to the output with the reset switch (R) that lies in parallel to the feedback capacitor. The CSA output is connected to a source follower driving the output node (Out). The circuits mentioned so far have been designed for moderate speed of applications in low-radiation environments. For the CERN Large Hadron Collider

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(LHC), where the time difference between consecutive crossings of particle bunches (25 ns) is much shorter than the time it takes to decide whether or not the data of a particular event needs to be kept (approximately 2 μs), chips with high-speed operation and radiation hardness have been developed successfully. In addition to fast low-noise amplifiers and radiation hardness, it is required to store the information for approximately hundred bunch crossings. The task of designing radiation hard electronics has been considerably eased by the industrial development of submicron integrated circuit technology which, due to the use of ultra-thin oxide, to a large extend has eliminated the problem of radiation induced threshold shifts in MOS transistors [17]. Taking some precautions in the design these technologies can be considered “intrinsically radiation hard”.

5.8 Silicon Drift Detectors The semiconductor drift detector was invented by E. Gatti and P. Rehak [1]. First satisfactorily working devices in silicon were realised in a collaborative effort by J. Kemmer at the Technical University Munich, the Max Planck Institute for Physics in Munich and the inventors [18]. The working principle may be explained by starting from the diode (Figs. 5.1 and 5.21a) if one realizes that the ohmic n+ contact does not have to extend over the full area of one wafer side but can instead be placed anywhere on the undepleted conducting bulk (Fig. 5.21b). Then there is space to put diodes on both sides of the wafer (Fig. 5.21c). At small voltages applied to the n+ electrode, there are two space-charge regions separated by the conducting undepleted bulk region (hatched in Fig. 5.21). At sufficiently high voltages (Fig. 5.21d) the two space-charge regions will touch each other and the conductive bulk region will retract towards the vicinity of the n+ electrode. Thus it is possible to obtain a potential valley for electrons in which thermally or otherwise generated electrons assemble and move by diffusion only, until they eventually reach the n+ electrode (anode), while holes are drifting rapidly in the electric field towards the p+ electrodes. Based on this double-diode structure the concept of the drift detector is realised by adding an additional electric field component parallel to the surface of the wafer in order to provide for a drift of electrons in the valley towards the anode. This can be accomplished by dividing the diodes into strips and applying a graded potential to these strips on both sides of the wafer (Fig. 5.5). Other drift field configurations (e.g. radial drift) can be obtained by suitable shapes of the electrodes. Drift chambers may be used for position and/or energy measurement of ionizing radiation. In the first case the position is determined from the drift time. Furthermore, segmenting the n+ -strip anode in Fig. 5.5 into pads, a two-dimensional position measurement is achieved. Due to the small capacitive load of the readout electrode to the readout amplifier, drift detectors are well suited for high precision energy measurement.

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Fig. 5.21 Basic structures leading towards the drift detector: diode partially depleted (a); diode with depletion from the side (b); double diode partially depleted (c); double diode completely depleted (d)

5.8.1 Linear Drift Devices Although linear devices seem to be the most straightforward realisation of the drift detector principle, one encounters some nontrivial problems. They are due to the finite length of the biasing strips and the increasing potential to be applied to these strips, which leads to a very large voltage of several hundred or (for very large drift length) a few thousand volts. Therefore guard structures have to be implemented which provide a controlled transition from the high voltage to the non-depleted region at the edges of the device. A schematic drawing of the first operational silicon drift detector [18] is shown in Fig. 5.22. Anodes placed on the left and right side of the drift region collect the signal electrons generated by the ionizing radiation. The most negative potential is applied to the field-shaping electrode in the centre. Electrons created to the left (right) of this electrode will drift to the left (right) anode. The p+ -doped field electrodes do not simply end on the side, but some of them are connected to the symmetrical strip on the other half of the detector. In this way one insures that the high negative potential of the field strips drops in a controlled manner towards the potential of the undepleted bulk on the rim of the detector.

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Fig. 5.22 Schematic cross-section and top view of a linear drift detector with p-doped fieldshaping electrodes (light) and two n-doped (double) anodes (dark)

Looking closely at the anodes (Fig. 5.22), it can be seen that there are pairs of ndoped strips. Each pair is surrounded by a p-doped ring, which also functions as the field-shaping electrode closest to the anode. The two n-doped strips are separated by a p-doped strip that also connects to the ring surrounding the anode. Surrounding the n-strips completely by p-doped regions ensures that the adjacent n-doped anodes are electrically disconnected to each other and to the other regions of the detector (such as the non-depleted bulk). The outer n-strips are used to drain away electrons from the high voltage protection region, while the inner strips measure the signals created in the active detector region. The opposite side of the silicon wafer is for the large part identically structured. Differences are only in the anode region, where the n-implantation is replaced by p-doped strips. In the main part of the detector, the strips on opposite sides of the wafers are kept at the same potential, thus assuring a symmetrical parabolic potential distribution across the wafer (Fig. 5.23a). Near the anode an increasing potential difference between the two wafer surfaces moves the potential valley for electrons to the front side until it ends at the anode (Fig. 5.23b). The linear drift detectors described so far allow one dimensional position measurement only. Dividing the anode of a linear drift detector into pads (Fig. 5.24) leads to a two-dimensional position measurement. One coordinate is obtained from the drift time, the other from the pads on which the signals appear. The second coordinate may be further improved by interpolation using the signal in neighbouring pads. The signal will be distributed over more than one pad if the

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Fig. 5.23 Electron-potential distribution in the linear region (a) and close to the anode region where the potential valley is directed towards the surface (b)

Fig. 5.24 Two-dimensional drift detector with the anode strip divided into pads. The dark pad anodes are embedded in a p-doped grid that provides insulation between neighbouring pads

diffusion during the drift time leads to a charge cloud at the anode that is comparable to the spacing of the pads. For very long drift distances and/or low drift fields, the signal charge will be spread over more than two readout pads. This is an undesirable feature when measuring closely spaced signals. Lateral diffusion can be suppressed by creating deep strip-like p-implanted regions parallel to the nominal drift direction [19]. In this way deviations from the nominal drift direction due to non-uniform doping of the silicon are also avoided.

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5.8.2 Radial and Single Side Structured Drift Devices Radial drift devices are in some sense simpler to design than linear devices because the problem of proper termination of the field-shaping strips does not occur. Radial devices are especially interesting for energy measurement. A small point-like anode with extremely small capacitance may be placed into the centre of the device. The small capacitance results in low electronic noise and as a consequence very good energy resolution. In one special case radial drift to the outside has been realized with a circular anode divided into pads, thus arriving at two-dimensional position measurement in cylindrical coordinates. An interesting feature of such an arrangement is the high position accuracy at small radius in the azimuthal direction. The position in this second coordinate is obtained from the charge distribution measured in the anode pads by projecting it back in the radial direction. A large-area device of this type [20], with a hole in the centre for the passage of the particle beam, has been produced for the CERES particle physics experiment at CERN. The device also uses a method to drain the current generated at the oxide-silicon interface between the field-shaping rings to an n-doped drain contact, separated from the signal-collecting anode [21]. In this manner the anode leakage current is reduced and the measurement precision increased. The Silicon Drift Diode (SDD) [3] combines radial drift with a homogeneous unstructured backside radiation entrance window (Fig. 5.25). Its principal field of application is in (X-ray) spectroscopy where excellent energy resolution is required. A further significant improvement was obtained by integrating a readout transistor into the device (Fig. 5.26). In contrast to the original drift chamber with the electron potential valley located parallel to the wafer surfaces now only one structured surface provides the drift field in the valley which now is at an angle with respect to the wafer surface.

Fig. 5.25 Cylindrical silicon drift detector. The entire silicon wafer is sensitive to radiation. Electrons are guided by an electric field to the small collecting anode in the centre

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Fig. 5.26 On-chip single sided junction FET coupled to the readout node of a cylindrical silicon drift detector

Having only one surface structured allows using the unstructured surface of the fully depleted device as radiation entrance window. Not having to take other functions into considerations, this radiation entrance window can be made very thin and uniform [22]. The circular geometry with a very small charge-collecting anode in its centre reduces the capacitive load to the amplifier and therefore the noise. Having the first transistor integrated into the device [23], the capacitance of the detector-amplifier system is minimized by eliminating bond wires between detector and amplifier. In this way stray capacitances between the readout node and ground are avoided, which makes the system faster and less noisy. Further advantages are evident as electrical pickup is significantly reduced and microphony i.e. noise introduced by mechanical vibrations, is excluded. In order to work on the lowly doped and fully depleted substrate, a non-standard “Single Sided Junction Field Effect Transistor” (SSJFET) has been developed [24]. Drift detectors with an integrated transistor are commercially available. They can also be obtained as modules assembled with a Peltier cooler in a gas-tight housing with a thin radiation entrance window (Fig. 5.27). To demonstrate the excellent spectroscopic performance achieved with such devices a spectrum obtained with an 55 Fe source and the quantum efficiency are presented in Fig. 5.28 for a cylindrical SDD with a sensitive area of 5 mm2 . The detector temperature, important for the leakage current, was set to −20 ◦ C and the signal shaping time to 1 μs. The MnKα line at 5.9 keV and the MnKβ line at 6.5 keV are clearly separated and their widths are only slightly above the intrinsic Fano limit given by the pair generation process in silicon. Cylindrical silicon drift diodes with integrated SSJFETs have been manufactured with sensitive areas in the range from 5 mm2 to 1 cm2 .

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Fig. 5.27 Perspective view of a module consisting of a single-sided structured cylindrical drift detector with integrated SSJFET transistor, cooled by a Peltier element

Fig. 5.28 MnKα – MnKβ spectrum (left) and quantum efficiency as function of X-ray energy (right) of a 5 mm2 drift diode. The device was operated at −20 ◦ C with a shaping time of 1 μs

5.9 Charge Coupled Devices Charge coupled devices (CCDs) have for a long time been used as optical sensors, most noticeably as imaging devices in video cameras. Some years ago they also found their application as particle detectors in Particle Physics [25], where specially selected optical CCDs were used. Meanwhile detector systems have been constructed for measuring tracks in electron-positron collisions [26]. p-n CCDs for the special purpose of particle and X-ray detection have been developed [2]. They are based on the principle of side-wards depletion of a double-

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diode structure, which is also used in the semiconductor drift chamber. Their first use was in two space-based X-ray telescopes: XMM [27] and ABRIXAS [28]. CCDs are non-equilibrium detectors. Signal charge is stored in potential pockets within a space-charge region, the content of which is then transferred to a collecting readout electrode. In order to retain the thermal non-equilibrium condition, thermally generated charge that also assembles in the potential pockets has to be removed from time to time. Usually this is done during the readout cycle of the device. While in conventional MOS CCDs minority carriers (electrons in a p-type bulk) are collected, the p-n CCDs are majority carrier (electrons in an n-type bulk) devices. The conventional MOS CCDs to be described in the following for didactic purposes store and transfer the charge directly at the semiconductorinsulator interface. These devices are in practice not used anymore and have been replaced by buried-channel CCDs, in which the store-and-transfer region is moved a small distance away from the surface. As a result they are less sensitive to surface radiation damage. In p-n CCDs, this region is moved a considerable distance into the bulk.

5.9.1 MOS CCDs The CCD transfer mechanism is explained in Fig. 5.29 that shows a cut along the transfer channel. The top part of the p-type bulk is depleted of charge carriers and the potential along the Si-SiO2 interface is modulated in a periodic fashion with the help of the metal electrodes on top of the SiO2 . Electrons created in the sensitive bulk region assemble in the potential maxima (minima for electrons) at the Si-SiO2 interface. The charge can now be moved towards the readout electrode by a periodic change of the voltages φ 1 , φ 2 , and φ 3 , as shown in the figure. First φ 2 is increased to the same level as φ 1 and the signal charge will spread between φ 1 and φ 2 . If now φ 1 is lowered, the signal charge will transfer below the electrodes φ 2 . If this procedure is followed for φ 2 and φ 3 and then again for φ 3 and φ 1 , the signal charge is transferred by a complete cell. After several cycles the charge will finally arrive at the anode, where it can be measured. Placing many of these channels next to each other and separating them by so called channel stops one arrives at a matrix CCD. Channel stops prevent the spreading of signal charge to neighbour channels. They can be realized by doping variations as for example an increased p-doping between channels. Usually charge is transferred into one additional charge transfer channel oriented perpendicular to the matrix channel (Fig. 5.30) so that the pixel charge can be shifted towards a single output node.

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Fig. 5.29 Working principle of a three-phase MOS CCD: layout (a); charge-transfer (b): Every third gate electrode is connected to the same potential (φ 1 , φ 2 , φ 3 ) so that a periodic potential appears below the gates at the Si-SiO2 interface. Electrons are collected in the maxima of the potential distribution. They can be shifted towards the readout anode by changing the potentials, as shown in (b)

Fig. 5.30 Matrix CCD and the principle of the charge-transfer sequence. Charge is shifted in the vertical direction with all pixels of the matrix in parallel, the lowest row being transferred into a horizontal linear CCD. This horizontal CCD is then read out through a single output node

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5.9.2 Fully Depleted pn-CCDs pn-CCDs were originally developed for X-ray imaging in space. A 6 × 6 cm2 size device is used as focal imager in one of the three X-ray mirror telescopes at the European XMM/Newton X-ray observatory [29]. From 2000 until the end of the mission in 2018 it has produced high quality X-ray images of the sky [30]. The pn-CCD principle, derived from the silicon drift chamber, has already been shown in Fig. 5.5. The layout of the XMM focal plane detector is shown in Fig. 5.31. Twelve 1 × 3 cm2 CCDs with 150 × 150 μm2 pixel size are monolithically integrated into a single device placed on a 4 inch silicon wafer of 300 μm thickness. Each column of pixels has its own readout channel allowing for fast parallel readout. Figure 5.32 shows a cross section of a pn-CCD along the transfer channel. Here one sees in greater detail the functioning of the device. Contrary to standard MOS-CCDs the registers are formed as pn-junctions and the radiation sensitive oxide plays only a minor role. The device is fully depleted with a higher n-type doping concentration in the epitaxial layer below the top surface. This leads to a potential distribution shown in the right part of the figure and prevents holes from the p+ -doped registers to be emitted across the wafer towards the backside p-doped entrance window. Charge storage and transfer occurs in a depth of approximately 10 μm in contrast to MOS CCDs where this happens at the Si-SiO2 interface. Fast and efficient charge transfer by drift is therefore possible even for large pixel sizes.

Fig. 5.31 Layout of the XMM pn-CCD. 12 logically separate pn-CCDs of 1 × 3 cm2 area are monolithically fabricated on a 4 in. wafer to a 6 × 6 cm2 device with a common backside entrance window. The pixel size is 150 × 150 μm2

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Fig. 5.32 Cross section through the CCD along the transfer channel

5.9.3 CCD Applications MOS CCDs have a long history in optical imaging. They have been used in camcorders but also in optical astronomy. In particle physics they were first used by the ACCMOR collaboration in the NA11 experiment at CERN where they were successfully employed for heavy flavour decay detection and measurement. They then found their way to collider physics at SLAC and also to X-ray astronomy, where thinning for backside illumination was necessary to achieve sensitivity for low energy X-rays. Thinning reduces the sensitive volume and therefore the sensitivity at higher X-ray energies. This disadvantage is avoided with pn-CCDs that have a typical thickness of 500 μm and, in addition are built with a ultra-thin entrance window so that high quantum efficiency at both low (100 eV) and high (20 keV) X-ray energies is reached. Good radiation tolerance for X-rays is due to two reasons, the absence of sensitive MOS registers and the absorption of X-rays within the bulk before they reach the sensitive charge transfer region (self-shielding). At XMM/Newton pnCCDs have been operating in space for 18 years without noticeable performance degradation. Compared to MOS CCDs the readout speed is significantly increased due to the larger pixel size, the higher charge transfer speed and parallel column readout. Very large pixel sizes cannot be realized in MOS CCDs that transfer charges very close to the Si-SiO2 interface. Use in a further X-ray mission is in preparation: eROSITA (extended ROentgen Survey with an Imaging Telescope Array). Here the CCD is split into an image collecting area and a frame store area. After collection, the complete image is transferred very fast into the frame store area from where it is read with moderate speed row by row while at the same time the next image is collected. The typical image frame readout takes 1 ms, while for MOS CCDs it is in the range of 1 s.

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Fig. 5.33 Schematic section through the CAMP detector. The reaction electron and ion detectors with the first CCD sensor plane are depicted on the left hand side. The pn-CCD detectors shown in perspective view on the right can detect all photons emerging from the target. In addition, the design allows feeding in other lasers for alignment or pump-probe purposes, as well as for mounting other high-resolution, small-solid-angle electron TOF or crystal spectrometers. The pnCCD1 can be moved in all three directions with a maximum distance of 25 cm along the beam trajectory

Although pn-CCDs have been developed for X-ray astronomy they are also visible-light detectors. One application is in adaptive optics that corrects in real time mirror geometries of optical telescopes in order to compensate for atmospheric turbulences at frequencies of approximately 1 kHz. pn-CCDs are also used in experiments at accelerator-based light sources in particular at X-ray Free Electron Lasers (e.g. FLASH and the European XFEL at Hamburg and LCLS at SLAC). The Center of Free Electron Science (CFEL) in Hamburg has designed the CFEL-ASG Multi Purpose (CAMP) chamber (Fig. 5.33) [31], which combines electron and ion momentum imaging spectrometers with large area, broadband (50 eV to 25 keV), high dynamic range, single photon counting and imaging X-ray detectors based on pn-CCDs. The excellent low energy response of pn-CCDs has been demonstrated by measuring the response to 90 eV photons at FLASH (Fig. 5.34).

5.10 Active Pixel Detectors The CCDs discussed in the previous chapter collect charges in pixels during their charge collection period and transport them during the transfer period pixel by pixel to a readout node. Charges produced during the transfer cycle will also be read but the assigned position will be wrong. In active pixel detectors each pixel has its own readout channel and the charge will be assigned to the pixel where it was generated. There are four types of active pixel detectors:

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Fig. 5.34 Energy resolution measured at FLASH with 90 eV photons. Every photon generates approximately 25 electron-hole pairs, which are detected with a read-out noise of 2.5 electrons (rms). The measured FWHM energy resolution is only 38.9 eV

(a) Hybrid pixel detectors are diode arrays bonded to an electronics chip produced on a separate wafer so that each pixel has its own readout channel. (b) MAPS (Monolithic Active Pixel Sensors) are pixel arrays with readout for every pixel directly integrated on the same chip. (c) DEPFET pixel detectors are two dimensional arrays of DEPFETs with parallel charge collection in the DEPFETs and serial delayed readout of the charges stored in the internal gates. (d) DEPFET Macro Pixel detectors, pixel detectors with large cell size combine DEPFETs with drift detectors. All these detector types exist in many variations. Hybrid pixel detectors and MAPS allow parallel data processing and can perform complex tasks thanks to the miniaturized VLSI electronics. This however has a price in power consumption. DEPFET pixel detectors so far are built in a technology of moderately large feature size. Thus complex data processing is not foreseen. Its advantages are sensitivity over the whole bulk, high energy resolution and very low power consumption.

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5.10.1 Hybrid Pixel Detectors Hybrid pixel detectors are used at the Large Hadron Collider (LHC) as the tracking detectors closest to the beam, where the track densities is highest and the radiation exposure most severe. They also became a standard detector for X-ray imaging, in particular at accelerator driven X-ray sources. In their simplest form they consist of a detector wafer with a two dimensional diode array and separate electronics wafers as shown in Fig. 5.35. Every diode is individually connected by bump bonding to its own readout channel. Other connection techniques, including capacitive coupling, have been demonstrated. As readout and sensor are separate, the sensor material can be freely chosen, e.g. a high-Z sensor for the detection of high-energy X-rays. The main challenge in such a device lies in the electronics that has to provide several functions as for example low noise charge readout and high dynamic range, and—depending on the application—data storage, zero suppression and transmission to the external electronics in analogue or digital form. These functions have to be implemented on an area of the pixel size. Frequently very high speed operation at low power is required as is the case for example in the LHC at CERN. Reaching these goals has been possible by profiting from the dramatic industrial progress in submicron electronics and adapting it to the specific needs. The use of submicron electronics that uses very thin gate oxides has also alleviated the problems with respect to radiation damage. The typical pixel dimension for the hybrid pixel sensors presently operating at the CERN LHC are of order 100 × 100 μm2 . The modules of the ATLAS vertex

Fig. 5.35 Concept of a Hybrid Pixel Detector consisting of a diode array “flip chip” bonded to several readout chips

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Fig. 5.36 Photo of an ATLAS pixel detector module

detector, shown in Fig. 5.36, have a pixel size of 50 μm × 250 (400) μm, the ones of CMS 100 μm × 150 μm For the High-Luminosity LHC hybrid pixel detectors with pixel sizes of 50 μm × 50 μm and 25 μm × 100 μm are under development. The hybrid pixel detectors used for X-ray science face somewhat different challenges and follow different concepts. AGIPD (Adaptive Gain Integrating Pixel Detector) [32], which operates at the European XFEL at Hamburg, where X-rays are delivered in pulse-trains with 220 ns distance between pulses, is designed to detect single and up to 104 photons with energies in the range 5–15 keV per pulse in pixels of 200 μm × 200 μm, and store 350 frames to be read out in between the pulse trains. This is achieved by signal-driven switching into four gain ranges. In addition, the 500 μm thick pixel sensor is designed for a breakdown voltage above 900 V for ionizing doses up to 1 GGy. There are many applications in X-ray science, where the recording of individual frames is not required, but the number of hits above a given threshold or in a given energy interval are counted for every pixel or the integrated charge for a given time interval recorded. As the electronics takes significantly less space than required for recording and storing individual frames, pixel sizes as small as 55 μm × 55 μm have been achieved. Outstanding examples for such detectors are PILATUS [33] developed at PSI, and the MEDIPIX series [34], developed by a collaboration centred at CERN.

5.10.2 Monolithic Active Pixel Sensors (MAPS) This name is used for pixel sensors produced with integrated circuit technology on a single wafer using part of the substrate as detector material. One advantage of MAPS is the significantly easier fabrication of detector modules resulting in a significant cost reduction; another is that MAPS can be produced in CMOS Fabs, which includes a fast turn-around time for the development. However, MAPS are very complex devices and achieving all the requirements of the experiments at highluminosity, including their radiation performance remains a challenge.

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Fig. 5.37 Cross section through a pixel of a MAPS fabricated on CMOS technology but using only NMOS transistors

A first successful demonstration of MAPS operating in an experiment is the EUDET beam telescope [35], with MAPS using only n-channel transistors out of an original CMOS technology. Figure 5.37 shows the cross section through a MAPS pixel cell. The n-well is used as collecting electrode and all transistors are placed within the p-wells. A small volume next to the n-well is depleted of charge carriers. In this region signal electrons are collected by drift, but, the major part of the sensitive volume—the p-epitaxial layer—is field-free. Thus most of the charge is collected by diffusion, which is intrinsically slow and leads to a large spread of charge into neighbouring cells. There are good reasons why p-type transistors are avoided. They would have to be placed into an n-well. If this well were separated from the charge collecting electrode it—depending on the n-well potentials—would collect signal electrons in competition to the signal electrode or might even inject electrons into the bulk. If it were put into the same well as the collecting electrode it would induce charge directly into the input of the pixel. For photon detection—as shown in the figure—in addition the material on the top as for example the conducting leads as well as the thick insensitive well zones will absorb part of the incident radiation. The pixel circuitry (Fig. 5.38) is rather simple. It consists of an NMOS input transistor, a reset transistor and an output select switch. Signal charge is stored at the

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Fig. 5.38 Pixel circuitry of MAPS based on CMOS technology but using only three NMOS transistors. The collecting electrode is directly connected to the gate of a source follower (M2) whose load is common to all pixels of a column and activated by the column select switch. The input node is reset with the reset transistor M1

Fig. 5.39 DMAPS with large collection electrodes (figure from Wermes-Kolanoski)

Fig. 5.40 DMAPS with small collection electrodes

input node, read out sequentially and cleared afterwards. MAPS using both CMOS types have also been developed [36]. To overcome the problem of slow charge collection by diffusion, which also makes the sensor sensitive to bulk radiation damage, DMAPS (Depleted CMOS Active Pixel Sensors), are being developed [37]. They are fabricated on substrates with resistivity between 100 ·cm and a few k ·cm and operated with depletion depths of typically 50–200 μm. As shown in Figs. 5.39 and 5.40, two approaches

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are followed: Large Collection Electrode (a) and Small Collection Electrode (b). Design (a) has the advantage of a more uniform electric field resulting in shorter drift distances, and thus a good radiation tolerance is expected. Its disadvantage is the large capacitance of about 100 fF per pixel and an additional well-to-well capacitance of similar value, which results in increased noise, reduced speed, higher power consumption and possibly cross-talk between sensor and digital electronics. Design (b) has a small electrode adjacent to the well in which the electronics is embedded. This has the advantage of a small capacitance of about a few fF and thus improved noise and speed at low power. However, the electric field in the sensor is not uniform with low field regions. This makes them more sensitive to radiation damage. DMAPS of both types have been fabricated by different foundries in 150 nm, 180 nm and 350 nm technologies. They show impressive results even after irradiation with hadrons to fluences exceeding a few 1015 cm–2. .

5.10.3 DEPFET Active Pixel Sensors The Depleted Field Effect Transistor structure shown in Fig. 5.6 is a natural building element for a pixel detector. It acts simultaneously as detector and as amplifier. A variety of DEPFET designs can be constructed. Figure 5.41 shows two examples, one with cylindrical, the other with linear geometry. Arranging many of these devices in a matrix and connecting them in such a way that selected DEPFETs can be turned on, one arrives at a pixel detector with charge

Fig. 5.41 Schematic drawings of MOS-type DEPFETs with circular (left) and linear (right) geometry. The signal charge is collected in a potential well (“internal gate”) below the FET gate, thereby increasing the conductivity of and thus the current in the transistor channel. The collected charges can be drained towards the clear contact by applying voltage pulses to the clear contact and/or the clear gate

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storing capability. Before turning to the matrix arrangement the main properties of the DEPFETs are summarised: • Combined function of sensor and amplifier; • Full sensitivity the over complete wafer, low capacitance and low noise, nondestructive repeated readout, complete clearing of the signal charge and thus no reset noise. • Continuous (real time) and integrating (charge storage) operating modes can be chosen. The signal can be read out either at the source as indicated in the left figure or at the drain as shown in the linear example. With source readout one compensates the increase of channel conduction due to the charge in the internal gate by a reduction of the external gate-source voltage, seen as voltage change of the source. In the drain readout the source potential is kept constant and the drain-current change can be directly observed. An important property in pixel detector applications is the fact that the signal charge collection occurs not only for current carrying DEPFETs but also for those which have been turned off with the help of the external FET gate. DEPFET pixel sensors have been developed at the MPI Semiconductor Laboratory in Munich for several purposes, as focal sensors of the proposed European X-ray observatory XEUS [38] and as vertex detector for the BELLE-II experiment at KEK in Japan and the proposed International Linear Collider ILC. In XEUS the combined functions of imaging and spectroscopy are of importance, for the vertex detectors the measurement of position of charged tracks is of prime interest. This however has to be done with very high precision (few μm) and at high readout speed. The position measurement requirement in XEUS is not as stringent; it is matched to the expected quality of X-ray imaging. However, highest emphasis is given to spectroscopic quality and quantum efficiency and data readout speed is still large. As a consequence of these and further requirements circular geometries have been chosen for XEUS and linear ones for the vertex detectors (see Fig. 5.41). The excellent spectroscopic capabilities of DEPFETs can be appreciated from the 55 Fe source spectrum taken with a single circular pixel cell (Fig. 5.42). The DEPFET with its capability of creating, storing and amplifying signal charge is an ideal building block for a pixel detector. A large number of DEPFETs can be arranged in a matrix in such a way as to power selected DEPFETs for reading and clearing the collected signal charge. Figure 5.43 shows a rectangular arrangement of DEPFETs. Their drains are connected column wise while gates and clear electrodes are connected row wise. Each row has its individual readout channel. A row at a time is turned on with the help of the gate voltage while all other DEPFETs have zero current. Charge collection does not require a current within the DEPFET. Readout can be performed in double correlated mode: Turning on the current with a negative voltage on the gate is followed by a first reading of the current, a clearing of the signal charge in the internal gate with a positive pulse at the clear contact and a second current reading before the current is turned down again and reading is switched to the next row. The difference of first and second current

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Fig. 5.42 55 Fe spectrum measured with a single circular (XEUS-type) DEPFET. A spectral resolution of 131 eV has been obtained with room temperature operation and 6 μs Gaussian shaping. The separately measured noise peak has a FWHM of 19 eV corresponding to an electronic noise of 2.2 electrons r.m.s

Fig. 5.43 Circuit diagram of a DEPFET pixel detector with parallel row-wise readout of the drain current

reading is a measure for the signal charge in the pixel cell. Alternatively to the procedure described above, sources may be connected column wise and source voltages measured instead of drain currents. Figure 5.44 shows the spectroscopic quality reached with a 64 × 64 DEPFET matrix of 50 × 50 μm2 pixels.

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Fig. 5.44 55 Fe spectrum measured at −28 ◦ C with a 64 × 64 cell DEPFET pixel matrix with 50 μm pixel size

Pixel sensors with large pixels can be constructed by combining DEPFET structure and drift chamber principle. Large pixel may be preferred in order to increase the readout speed and reduce the number of readout channels and power consumption. It is advisable to match the pixel size to the properties of the rest of the system. Over-sampling may increase the electronic noise lead to a worse performance. Macro Pixel DEPFET Sensors Figure 5.45 shows the principle with a cut and a top view of a cell. The circular DEPFET structure is located in the centre of a cylindrical drift detector. Electrons created anywhere in the fully depleted bulk are driven by the suitably shaped drift field towards the internal gate below the transistor channel. For this device a new type of DEPFET has been invented that allows clearing of the signal charge with substantially lower voltage by putting the clear electrode inside the drain region located in the centre. The drain region does not consist of a highly doped p region but is formed by an inversion layer that is controlled by a gate voltage and automatically connected to the small drain contact. Putting a sufficiently high positive voltage on this gate, the drain assumes the role of the clear electrode, which is automatically connected to the n-doped clear contact. Single pixel cells and a 4 × 4 1 mm2 pixel matrix (Fig. 5.46) have been tested successfully. Figure 5.47 shows an 55 Fe spectrum taken at room temperature. Here one notices a somewhat worse spectroscopic resolution than with the small-pixel devices. This is due to the leakage current which now is collected from a volume which is larger by a factor 400. The leakage current can be suppressed by lowering the operating temperature.

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Fig. 5.45 Principle of a macro-pixel cell: A DEPFET located at the centre of a drift detector serves as storage and readout device

New DEPFET Developments The DEPFET concept allows a variety of further functionalities that have partially been proven experimentally but not yet implemented into a large area pixel detector: (a) As signal charge is not destroyed by the readout process this charge can be read repeatedly and the measurement precision improves with the square root of the number of measurements. This has been verified with a pair of neighbouring DEPFET transistors arranged in such a way as to allow the transfer of signal charge from one internal gate to the other and in reverse direction. A measurement precision of 0.25 electrons has been achieved independently of the amount of signal charge [39]. (b) Gatable DEPFETs [40] are developed for applications in High Time Resolution Astronomy (HTRA) and Adaptive Optics. They collect signals in preselected time intervals only, whereas the charge generated outside of these gate periods are drained towards a clear electrode.

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Fig. 5.46 Layout of a macro pixel matrix

Fig. 5.47 55 Fe spectrum measured at room temperature in a 1 × 1 mm2 pixel of an 8 × 8 macro pixel matrix with 6 μs shaping. The increase of the noise compared to single DEPFET cells is due to the leakage current in the large sensitive volume of 1 × 1 × 0.45 mm3 , which can be reduced by cooling

(c) Nonlinear DEPFETs [41] developed for applications at the European X-ray Free Electron Laser (EuXFEL) at Hamburg. Their non-linear characteristics and high-speed capability combines simultaneously single X-ray-photon sensitivity and very high dynamic range at the 5 MHz EuXFEL repetition rate.

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DEPFET Pixel Detector Applications In the last years DEPFET pixel detectors have been developed at the MPI Semiconductor Laboratory for the following projects: Bepi Colombo, a mission for observing mercury [42], XEUS/IXO a space based X-ray observatory that will succeed the XMM/Newton and vertex detectors for the International Linear Collider (ILC) and the BELLE-II experiment at the KEK e+ e– collider. As an example for the application in X-ray detection Fig. 5.48 shows spectra at high readout rates taken with a Bepi Colombo prototype macro pixel detector. In the final detectors (Fig. 5.49) the pixel size is reduced to 300 × 300 μm2 . An X-ray image obtained by illumination through a mask (Fig. 5.50) demonstrates functioning of the full detector. The ILC and BELLE vertex detectors [44, 45] require fast readout (10 μs frame time), excellent spatial resolution (5 μm) and minimal material thickness to

Fig. 5.48 Spectroscopic resolution of Bepi Colombo macro-pixel detectors with 64 × 64 pixels of 500 × 500 μm2 size on a 500 μm fully depleted substrate with ultra-thin backside radiation entrance window. The top figure is restricted to photons contained in single pixels. while in the lower part signals split between neighbour pixels are included. Readout was with the ASTEROID pixel chip [43] that averages the DEPFET signals over an “integration time” once before and once after clearing and takes their difference as a measure for the deposited charge. The measured width of 125 eV FWHM with 0.9 μs integration time corresponds to an electronic noise of 4 electrons r.m.s. Reducing the integration time from 0.9 to 0.25 μs increases the width to 163 eV FWHM corresponding to 13 electron charges r.m.s

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Fig. 5.49 Photo of an assembled macro-pixel detector with two 64 channel ASTEROID readout chips on top and bottom and four steering chips

Fig. 5.50 X-ray image (right) obtained with the mask shown on the left

minimize the scattering of charged particles. Consequently the pixel size has been chosen as 25 × 25 μm2 for ILC and 50 × 75 μm2 for BELLE-II. A new method for wafer thinning based on wafer bonding technique has been developed in order to produce thin (50 μm) self-supporting all silicon modules [46].

5.11 Detectors with Intrinsic Amplification Contrary to gas detectors, semiconductor detectors usually provide only the primary ionization as signal charge. This mode of operation is possible because of the low energy needed for producing an electron-hole pair (3.6 eV in silicon, whereas the ionization energy for gases is about 30 eV) and the availability of low noise

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electronics. The measurement of the primary ionization without gain avoids any effect of gain variation or amplification noise, and thus leads to stable operation in spectroscopic measurements. However, high speed and very low noise requirements, detection of single photons, compensation for charge losses due to radiation damage or timing accuracies of the order of tens of picoseconds, make an intrinsic amplification of the detectors desirable. A rather old and well known device is the avalanche diode, with several different operating modes. In the last two decades arrays of avalanche diodes operated in the Geiger mode (SiPMs—Silicon Photo Multipliers) have become photo-detectors of choice for many applications, and more recently tracking detectors with gain (LGAD—Low Gain Avalanche Detectors) are developed with the aim to combine precision position with precision timing in the harsh radiation environment of the high-luminosity LHC at CERN.

5.11.1 Avalanche Diode An avalanche diode has a region with a field of sufficient strength to cause charge multiplication. An example of such a device is shown in Fig. 5.51. The base material is low doped p-type silicon. The junction, consisting of a thin highly doped n-type layer on top of a moderately doped p-layer, may also be used as entrance window for radiation, especially when the bulk material is only partially depleted. An enlarged view of the central top region of Fig. 5.51, in which multiplication takes place, is shown in Fig. 5.52. Also shown are charge density, electric field and potential for the idealized assumption of uniform doping in the n+ -, p- and p− -regions ignoring diffusion. The middle p region is fully depleted and the spacecharge region extends into the thin n+ top region and the low doped p− -bulk. The maximum of the electric field is at the n+ p junction. Electrons produced below the n+ p junction (and holes produced above the junction) will pass the high field region of the junction when drifting in the electric

Fig. 5.51 Avalanche diode built on p-type silicon with a high-field region right below the top surface

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Fig. 5.52 Amplification region of the avalanche diode shown in Fig. 5.51. Also shown are charge density ρ, electric field E, and potential V

field towards the collecting electrode on top (on bottom). If the electric field is strong enough to accelerate electrons (or holes) between collisions with the lattice imperfections so that the kinetic energy is sufficient to create another electron-hole pair, the charge produced by the primary ionization is amplified. One important aspect to be considered in designing or operating avalanche diodes is the different behaviour of electrons and holes with respect to charge multiplication. In silicon, the onset of charge amplification for holes occurs at higher electric fields than for electrons. The situation is opposite in germanium, while in GaAs the difference between electrons and holes is comparatively small. Therefore several working regimes exist that vary depending on the strength and extension of the high electric field region. In the case of silicon one finds: (a) At low electric field, no secondary electron-hole pairs are generated. The device has the characteristics of a simple diode. (b) At higher electric field only electrons generate secondary electron-hole pairs. The amplified signal will be proportional to the primary ionization signal, with some statistical fluctuation from the multiplication process added to the fluctuation in the primary ionization process. (c) At even higher field, holes will also start to generate secondary electronhole pairs. Secondary electrons generated by holes will again pass through (part of) the amplification region, thereby possibly generating other (tertiary) electronhole pairs. This avalanche process will continue until it is either stopped by a statistical fluctuation in the multiplication process or by a sufficiently large drop of the externally supplied voltage. This drop may be due to the increased current passing through a bias resistor or an external enforcement by, for example, a feedback circuit. The generation of a large number of free charge carriers in the multiplication region also reduces the electric field strength and therefore decreases charge multiplication in later stages of the avalanche generation. In this operation mode the output signal is no more proportional to the primary charge; however, single photon detection becomes possible.

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5.11.2 Low Intensity Light Detection An optical photon in its primary interaction will create a single electron-hole pair, a charge too small to be detected by standard electronics. However, intrinsic amplification in an avalanche process makes single photon detection possible. The avalanche diode of Fig. 5.51 is such a device. Operation in proportional mode will result in an output signal proportional to the number of (optical) photons, with some statistical fluctuations of the avalanche process added and additional contributions from the non-uniformity of the electric field in the avalanche region. Operation in limited Geiger mode will result in a signal independent of the number of incident photons. The charge signal will be approximately given by the product of the diode capacitance times the difference of the applied voltage and the voltage at which the avalanche process stops. As the charge multiplication probability is a strong function of the electric field strength, high uniformity over the active area is required and high field regions at the edge of the device have to be avoided by proper design. Edge breakdown is avoided in Fig. 5.51 by the less strongly doped n region at the rim. This leads to a spacecharge region extending deeper into the bulk and to a reduction of the maximum field. If the structure of Fig. 5.51 is to be operated in proportional mode (with only electrons multiplying), primary charge produced by radiation entering from the top has to be generated below the high field multiplication region in order to be properly amplified. Therefore for blue light, with its submicron penetration depth, the efficiency is low for this design. In choosing the width of the depleted region, one has to consider several partially conflicting requirements. Based on noise considerations, this region should be large in order to reduce the capacitive load to the amplifier. The same is required for the detection of deeply penetrating radiation such as X-rays or energetic charged particles. One may even extend the depleted region all the way to the bottom surface. Then the backside p-doped surface can also be used as a radiation entrance window. This can be an advantage for low penetrating radiation such as optical photons, since such an entrance window can be made thin. The disadvantage of a large depleted region is the large volume for thermal generation of electron-hole pairs, the electrons being capable of initializing the avalanche process and, depending on the application, a not wanted sensitivity to deeply penetrating radiation. The electric field configuration in the avalanche region is shown in an idealized way in Fig. 5.52, assuming abrupt doping changes. Such a distribution is not only unrealistic but also far from optimal for proportional operation: Breakdown should be avoided as much as possible which can be achieved by an extended amplification region and lower hole-to-electron multiplication ratios, as is the case for lower fields. Such a design can be realised by suitably doping the avalanche region.

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5.11.3 Solid-State Photo Multipliers: SiPMs In the last decade a new type of avalanche photon detector has reached maturity and is now commercially available, the Solid State Photo Multiplier, also referred to as SiPM (Silicon Photo Multiplier), G-APD (Geiger Mode Avalanche Photo Diode) or MPPC (Multi Pixel Photon Counter) [47]. It consists of two dimensional arrays of 100–10,000 single photon avalanche diodes (SPADs), called pixels, with typical dimension between (10 μm)2 and (100 μm)2 . The pixels are operated in limiting Geiger mode and every pixel gives approximately the same signal, independent of the number of photons which have produced simultaneously electron-hole pairs in the amplification region of the pixel. The sum of the pixel signals is equal to the number of pixels with Geiger discharges, from which the number of incident photons can be determined. As the output charge for a single Geiger discharge is typically larger than 105 elementary charges, 0, 1, 2, and more Geiger discharges can be easily distinguished, enabling the detection of single optical photons with high efficiency and sub-nanosecond timing. The quenching of the Geiger discharge is either achieved by a resistor in series with each pixel or an active feedback. Two types of SiPMs have been developed: Analogue and Digital. In Analogue SiPMs [47] the individual pixels are connected to a common readout and the SiPM delivers the summed analogue signal. In Digital SiPMs [48] each pixel has its own digital switch to a multi-channel readout system and the output is the digitized pulse height and precise time information for the pixels with Geiger discharges. Digital SiPMs also allow disabling pixels with high dark-count rates. The pulse shape and the gain of SiPMs are explained with the help of Figs. 5.53 and 5.54: A schematic cross section of a single pixel is shown in Fig. 5.53, and an electrical model of a pixel with resistor quenching, in Fig. 5.54. The bias voltage is denoted Vbias, the single pixel capacitance Cpix , and the quenching resistance Rq . Frequently, in particular for SiPMs with larger pixel sizes, a capacitance Cq parallel to Rq is implemented. In the quiescent state the voltage over Cpix is Vbias. When an electron-hole pair in the amplification region starts a Geiger discharge, in the model the switch is closed and Cpix is discharged through the current source until the turnoff voltage Voff is reached, at which the Geiger discharge stops and the switch opens. The assumption of a constant current source is certainly oversimplified. However the sub-nanosecond discharge time is so short, that details of the time dependence of the discharge current hardly affect the results of the simulation. If a finite capacitance Cq is present, a fast pulse with charge Cq ·(Vbias – Voff ) appears. After the switch opens, Cpix is charged up to Vbias with the time constant τ ≈ Rq ·Cpix and the total signal charge is approximately (Cpix + Cq )·(Vbias – Voff ). Figures 5.55 and 5.56 show two examples of pulse shapes: (a) For a KETEK SiPM with (15 μm)2 pixels and negligible Cq , and (b) for a KETEK SiPM with similar doping profiles however with (50 μm)2 pixels and a finite Cq . The value of Rq has to be sufficiently high to quench the Geiger discharge. As Cpix increases with increasing pixel area, τ = Rq ·Cpix also increases, and a finite Cq has to be introduced to achieve a good timing performance and an increased pulse height if fast pulse shaping is used.

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Fig. 5.53 Example of the schematic layout of a SiPM pixel. The Geiger breakdown occurs in the high-field n+ region, which has a depth of order 1–2 μm. The p++ -electrode of every pixel is connected through the quenching resistance (Rq ) to the biasing lines (Al) to which the biasing voltage Vbias is applied The photons enter through the transparent p++ -electrode

Fig. 5.54 Electrical model of a single SiPM pixel

Fig. 5.55 Pulse shape from a single photon for a KETEK SiPM with 4384 pixels of (15 μm)2 : A single exponential with τ = Rq ·Cpix ≈ 20 ns

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Fig. 5.56 Single photon pulse shape for a KETEK SiPM with 400 pixels of (50 μm)2 : A prompt signal due to the finite value of Cq and a slow component with the time constant τ = Rq ·Cpix ≈ 110 ns is observed

In our discussion we distinguish between the breakdown voltage Vbd , the threshold voltage for a Geiger discharge, and the turn-off voltage Voff , the voltage at which the Geiger discharge stops. Differences Vbd – Voff of up to about 1 V have been observed [49]. They should be taken into account when characterising or modelling SiPMs. We note that Vbd can be obtained from I–V measurements, as the voltage at which the current rises quickly due to the onset of Geiger discharges or the voltage at which the photon detection efficiency starts to differ from zero, and Voff can be determined from the dependence of SiPM Gain on Vbias by extrapolating the linear Gain(Vbias) dependence to Gain = 1. One outstanding feature of SiPMs is the single-photon resolution, as demonstrated in the charge spectrum shown in Figs. 5.57 and 5.58 [50]. 0, 1, . . . up to >30 simultaneous Geiger discharges can be distinguished allowing for straight-forward

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calibration methods. The high photon-detection efficiency, where after careful optimisation values in excess of 60 % for wavelengths between 250 and 600 nm have been reached, the high gain of typically 106 , and the intrinsic timing resolution of a few picoseconds, are other attractive performance parameters. In addition, SiPMs are not affected by magnetic fields, operate in a wide temperature range, are very robust, and work at moderate bias voltages (≈ 25–75 V). Also, thanks to the microelectronics technology, SiPMs have highly reproducible performance parameters and are relatively inexpensive. Limitations of SiPMs are their size, which is typically below 1 cm2 , and their limited dynamic range, essentially determined by the number of pixels. In addition, the measurement of the number of photons is affected by two sources of excess noise, which worsen the resolution beyond Poisson statistics: After-pulsing and Cross-talk. After-pulses are the result of charge carriers which are produced in the Geiger discharge and trapped in defect states. Depending on the energy in the silicon band gap and the properties of the defect states, they are released with different detrapping time constants and cause additional signal fluctuations, which depend on the integration time of the readout electronics. In Figs. 5.57 and 5.58, which show pulse-height spectra recorded with a 100 ns gate at room temperature, after-pulses can be seen as entries in-between the peaks. Cross-talk is produced by the photons from the accelerated charges in the Geiger discharge, which generate electron-hole pairs in adjacent SiPM pixels. The photon path can be inside of the silicon but also via reflection in the protective layer of the SiPM or a light guide. This light path is so short that this cross-talk can be considered as prompt. Implementing trenches filled with absorbing material in-between the pixels reduces the prompt cross-talk significantly. The photons from the Geiger discharge can also generate electron-hole pairs in the non-depleted region of the SiPM, which can diffuse into

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the amplification region and cause delayed cross-talk. The result of prompt crosstalk is that the number of entries in the peaks does not follow a Poisson distribution, even if the number of photons causing initial Geiger discharges does. As shown in [51] the result of cross-talk is that the number of entries in the peaks follows a Generalised-Poisson instead of a Poisson distribution. We note that the solid curve shown in Figs. 5.57 and 5.58 is the result of a model fit which includes both after-pulsing and prompt cross-talk simulated by a Generalised Poisson distribution. The model provides a fair description of the measurements and gives a precise determination of the SiPM parameters [50]. As both, after-pulses and cross-talk are related to the number of charge carriers in the Geiger discharge and thus to the Gain, the corresponding probabilities are expected to be approximately proportional to Vbias – Voff , which is also observed. Typical values at Vbias – Voff = 5 V for afterpulsing as well as prompt cross-talk are 5 % resulting in an excess noise factor, the ratio of the square of the relative resolution to the Poisson expectation, ENF = [(σmeas /meanmeas )/(σPoisson/meanPoisson)]2 of ≈ 1.08. As the photon detection efficiency increases with voltage and finally saturates, whereas Gain and ENF continue to increase, there is a voltage at which the photon number measurement is optimal. Dark counts are another limitation of SiPMs. Typical dark count rates (DCR) for SiPMs before irradiation are between 10 and 100 kHz/mm2 at room temperature. Cooling reduces the DCR by about a factor 2 for an 8 ◦ C reduction in temperature. Ionizing radiation, which mainly causes damage to the SiO2 , hardly affects the DCR. However non-ionizing radiation, like neutrons or high energy (> 5 MeV) particles, significantly affect the performance. At sufficiently high fluences () the DCR is so high that most pixels are in a state of Geiger discharge, the photon-detection efficiency decreases and finally the SiPM stops working as a photo-detector. Whereas Vbd and the electrical SiPM parameters hardly change up to  = 5 × 1013 cm–2 , DCR increases by many orders of magnitude: For a KETEK SiPM with 15 μm pitch at –30 ◦ C and (Vbias – Voff ) = 5 V, DCR increases from ≈ 10 kHz/mm2 before irradiation to ≈ 200 GHz/mm2 after irradiation by reactor neutrons to  = 5 × 1013 cm–2 [52, 53]. It is found that the increase in DCR is approximately proportional to . It is also observed that after irradiation the increase of DCR with excess voltage is significantly steeper and the decrease with temperature slower after than before irradiation. As a result of the increased DCR, the signal baseline shows large fluctuations and single photon detection becomes impossible. Finally the occupancy of the pixels by dark counts is so high that the probability of a photon hitting a pixel which is already busy increases and the photon detection efficiency degrades. For the KETEK SiPM with 15 μm pitch at – 30 ◦ C the photon detection efficiency due to dark counts is reduced by a factor 2 for  = 5 × 1013 cm–2 at (Vbias – Voff ) ≈ 2.5 V, and essentially zero for  = 5 × 1014 cm–2 [53]. At these high fluences the dark currents exceed several tens of mA and thermal run-away has to be avoided. After irradiation a significant reduction of DCR by annealing occurs. Annealing is a strong function of temperature: The typical reduction of DCR is a factor 2–3 after several days at room temperature, and a factor 10–50 at 175 ◦ C. A systematic

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study of different annealing scenarios, which allows to optimise the temperature cycling for operating SiPMs in high radiation fields, as available for silicon tracking detectors without gain [7, 10], is so far not available. In [54] it is demonstrated that SiPMs produced by Hamamatsu and SENSL, after irradiation to a fluence of 1014 cm–2 and annealed at 175 ◦ C can achieve single photon detection at 77 K with a DCR below 1 kHz/cm2. The values of Vbd and Voff have a temperature dependence of order 20 mV/◦ C, which results in a temperature-dependent gain. However this is not a real problem and several feedback systems for gain stabilisations have been designed and are used. Due to the vast application potential, which spans from research, over industrial applications to medicine, several firms develop and manufacture SiPMs. In close collaboration with research institutions, in particular working in particle physics, a rapid development and major improvements of SiPMs are presently under way.

5.11.4 Ultrafast Tracking Detectors: LGADs At the HL-LHC (High-Luminosity Large Hadron Collider at CERN planned to start operation in 2026) in the large collider experiments ATLAS and CMS there will be on average ≈ 200 interactions with vertices distributed over ≈ 10 cm along the beam direction for every bunch crossing. For the complete kinematic reconstruction of the most interesting interactions in a bunch crossing, the information of the individual detector components has to be assigned to the correct interaction vertices. To illustrate the problem, Fig. 5.59 shows the reconstructed tracks extrapolated to the interaction region for a single bunch crossing with 50 interactions recorded in 2012. For a few vertices the interaction times, which are spread over ≈ ±200 ps, as obtained from a simulation, are given. For an efficient assignment of tracks to vertices, tracking detectors with high efficiency, 5 μm position resolution, 20 ps Fig. 5.59 Interaction times of a number of proton-proton vertices in a single bunch crossing with 50 interactions [55]. The data have been recorded by the CMS experiment in 2012. At the HL-LHC, the average number of interactions per bunch crossing is expected to be about 200

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timing accuracy and 25 ns pulse shaping, are required. From simulations [55] it is concluded that pixel sensors with 50 μm active thickness and a doping profile similar to the one shown for APDs in Fig. 5.52 and operated at a gain of ≈ 20 can reach the required performance. These detectors are called Low-Gain-Avalanche Detectors, LGADs. Different to optical photons which generate single electron-hole pairs, minimumionizing produce about 75 charge pairs per micro-meter and a high gain is not required. In addition to increasing fluctuations, high gain causes also practical difficulties and increases the shot noise from the dark current. Thin sensors have the additional advantage of smaller dark currents and a pulse rise time which increases with decreasing sensor thickness. The effects which influence the timing accuracy can be grouped in five categories: (1) Position-dependent fluctuations of the charge carriers produced by the charged particle to be measured, (2) excess noise of the amplification mechanism, (3) position dependent drift field and coupling of the of the drifting charges to the readout electrodes, (4) electronics noise, and (5) digitisation error of the time-todigital convertor. A major issue for LGADs is the control of the gain after irradiation. The change of the effective doping by dopant removal and defect states, and the decrease of the mobilities and amplification coefficients of electrons and holes due to radiation damage appear to present major problems. These are addressed in an extensive R&D program which started in 2012 and has already given first encouraging results.

5.12 Summary and Outlook Different concepts of solid silicon sensors and the electronics required for their readout have been described in this contribution. Although a detailed theoretical understanding of silicon devices had already been achieved in the 1960s, silicon detectors remained a niche application, used mainly in Nuclear Physics. This changed around 1980, when Josef Kemmer adapted the planar technology of microelectronics to sensor fabrication and the ACCMOR Collaboration demonstrated the reliable long-term operation and excellent physics performance of silicon strip detectors. Based on these results, many groups started to develop and use silicon detectors, and today there is hardly a particle physics experiment, which does not rely heavily on them. The areas covered by silicon detectors in the particle physics experiments increased from tens of cm2 to hundreds of m2 . Large areas of silicon detectors are even used on satellites for space experiments. In parallel to silicon detectors, the development of low-noise ASICs and connection technology started. They are required for reading out the more and more complex silicon sensors. In addition, a number of industrial producers, in closed collaboration with academia, developed and fabricated silicon sensors. Today silicon radiation detectors are a quite big market. Initially developed for Particle Physics, the use of silicon detectors spread into many different fields of science, medicine and industrial applications.

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Since 1980 several new detector concepts were proposed, realised and used for a variety of measurement tasks. Outstanding examples are drift detectors, fully depleted CCDs, DEPFETs, MAPSs 3-D sensors, APDs and SiPMs. The different devices have their advantages and shortcomings, but offer high-performance solutions for most measurement tasks. In recent years radiation damage for the use of silicon sensors at high flux or high luminosity colliders has become more and more of a concern. Whereas radiation damage by X-rays can be controlled by a proper sensor design, the question up to which fluence of high-energy radiation silicon detectors can be used is a field of intense research. Unfortunately other sensor materials, like crystalline diamond or GaAs seem not to be a solution. Defect engineering, by doping crystals with different impurities has resulted in some improvements. However, a breakthrough for high fluences could not be demonstrated. Therefore the only approach appears to optimise the sensor layout for radiation tolerance. The recipe followed are high fields and low charge collection distances. How far intrinsic amplification can help remains an open question. For the design optimisation, complex TCAD (Technology Computer-Aided Design) simulations are performed. In spite of some first successes, a major progress is still required. As far as the electronics, which is exposed to the same fluences, is concerned, the sub-micron technology with nano-meter dielectric layers resulted in a big step in radiation tolerance. For the future there is the strong hope that detectors can be fabricated which achieve the challenging performance parameters in the high radiation fields of the HL-LHC and future high-luminosity colliders. The field of solid state detectors will also profit very much from the ongoing industrial R&D efforts, in particular of 3-D integration technology and nano-electronics. Last but not least I very much hope that, like in the past, radically new ideas will come up and expand further the applications of solid state detectors.

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Chapter 6

Calorimetry Particle Detectors and Detector Systems C. W. Fabjan and D. Fournier

6.1 Introduction, Definitions In particle physics, calorimetry refers to the absorption of a particle and the transformation of its energy into a measurable signal related to the energy of the particle. In contrast to tracking a calorimetric measurement implies that the particle is completely absorbed and is thus no longer available for subsequent measurements. If the energy of the initial particle is much above the threshold of inelastic reactions between this particle and the detector medium, the energy loss process leads to a cascade of lower energy particles, in number commensurate with the incident energy. The charged particles in the shower ultimately lose their energy through the elementary processes mainly by ionization and atomic level excitation. The neutral components in the cascade (γ, n,..) contribute through processes described later in this section. The sum of the elementary losses builds up the calorimetric signal, which can be of ionization or of scintillation nature (or Cherenkov) or sometimes involve several types of response. While the definition of calorimetry applies to both the low energy case (no showering) and the high energy case (showering), this section deals mostly with the showering case. Examples of calorimetry without showering are discussed in Sect. 6.2.3.

C. W. Fabjan Austrian Academy of Sciences and University of Technology, Vienna, Austria e-mail: [email protected] D. Fournier () IJCLab, Université Paris-Saclay, CNRS/IN2P3, Orsay, France e-mail: [email protected] © The Author(s) 2020 C. W. Fabjan, H. Schopper (eds.), Particle Physics Reference Library, https://doi.org/10.1007/978-3-030-35318-6_6

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Only electromagnetic and strong interactions contribute to calorimetric signals, the weak (and gravitational) interaction being much too small to contribute. Particles with only weak (or gravitational interaction) will escape direct calorimetric detection. An exception are the neutrino detectors discussed in Sect. 6.4: statistically, when a very large number of neutrinos cross a detector, a tiny fraction of them will interact (weakly) with matter and will lead to particle production which can be measured by different methods, including calorimetry. The measurement of the energy of a particle is the primary goal of calorimetry. In addition, several other important quantities can be extracted, such as impact position and timing, particle direction and identification. These issues are considered in Sects. 6.4–6.6, before addressing specific examples in Sect. 6.7. In Sect. 6.2 the fundamentals of calorimetry are presented, followed by a discussion of signal formation obtained from the energy deposition (Sect. 6.3). In recent years, calorimetry in the ATLAS and CMS detectors at the LHC played an essential role in the discovery of the Higgs boson, announced in July 2012.

6.2 Calorimetry: Fundamental Phenomena Given the large differences between electromagnetic interactions and strong interactions, the following subsections start with electrons and photons, which have only electromagnetic interactions (see however the end of this section), before addressing the case of particles with strong interactions, also called hadrons. The case of muons is considered in a separate subsection.

6.2.1 Interactions of Electrons and Photons with Matter Several elementary interaction processes of the electrons with the medium contribute to the energy loss −dE of an electron of energy E after a path dx in a medium: Møller scattering, ionization and scattering off the nuclei of the medium: bremsstrahlung (Fig. 6.1). Electron-electron scattering is considered as ionization (Møller) if the energy lost is smaller (larger) than me c2 /2. It is customary to include Fig. 6.1 Photon radiation from electron interaction with a nucleus (A, Z)

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Fig. 6.2 Average energy loss of electrons in copper by ionization and bremsstrahlung. Two definitions of the critical energy (E and ε (Rossi)) are shown by arrows

in energy loss by ionization atomic excitations, some of which lead to light emission (scintillation). For positrons the Møller scattering is replaced by Bhabha scattering. The calculated average energy loss is shown in Fig. 6.2 for copper and the average fractional energy loss (−1/E dE/dx) is plotted in Fig. 6.3 for lead [1]. Figure 6.2 illustrates that the average energy lost by electrons (and positrons see Fig. 6.3) by ionization is almost independent of their incident energy (above ~1 MeV), with however a small logarithmic increase. For electrons [1, 2]. √ '  Z 1 −dE γme c2 β γ − 1 1  2 =k 1 − β ln √ + dx A β2 2 I 2     2γ − 1 1 γ−1 2  − ln2 + MeV/ g/cm2 2 2γ 16 γ

Fig. 6.3 Relative energy loss of electrons and positron in lead with the contributions of ionization, bremsstrahlung, Møller (e− ) and Bhabha (e+ ) scattering and positron annihilation

(6.1)

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and for positrons √

Z 1 dE γme c2 β γ − 1 =k √ − ln dx A β2  I 2  2    14 10 β 4 23 + + + − MeV/ g/cm2 2 3 24 γ + 1 (γ + 1) (γ + 1) (6.2) In these formula, A (Z) are the number of nucleons (protons) in the nuclei of the medium, I is the mean excitation energy of the medium—often approximated by 16 Z0.9 eV—the constant k = 4πNA re 2 me c2 = 0.3071 MeV/(g/cm2), NA the Avogadro 2 1 number and re = 4πε · me c2 = 2.818 10−15 m the classical radius of the electron. 0 e For positrons the annihilation with an electron of the medium has to be considered. The cross section of this process (σ an = Zπ r e 2 /γ for γ > > 1) decreases rapidly with increasing energy of the positron. At very low energy, the annihilation rate is:   (6.3) R = NZ π re 2 c s−1 , with N = ρ NA /A, the number of atoms per unit volume. This rate corresponds to a lifetime in lead of about 10−10 s [3]. Positron annihilation plays a key role in some technical applications (Positron Emission Tomography, Chap. 7). Figure 6.2 shows that the average energy loss by bremsstrahlung (photon emission in the electromagnetic field of a nucleus) increases almost linearly as a function of incident energy (meaning that the fractional energy loss is almost constant, as shown in Fig. 6.3). This is described by introducing the radiation length X0 defined by: −dE/E = dx/X0

(6.4)

It follows from the definition that X0 is the mean distance after which an electron has lost, by radiation, all but a fraction 1/e of its initial energy. X0 also has a simple meaning in terms of photon conversion (see below). While X0 should show a small increase at low energy corresponding to a small drop in the fractional energy loss visible in Fig. 6.3, it soon reaches a high-energy limit which has been calculated by Bethe and Heitler [3, 4] and more recently by Tsai [5] and tabulated by Dahl [1] for different materials. In the seminal book by Rossi [6] the formula for X0 , based on the Bethe–Heitler formalism reads: . /    cm2 g−1 1/X0 = 4 α (NA /A) Z (Z + 1) re 2 ln 183Z −1/3

(6.5)

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The Z2 term reflects the fact that the bremsstrahlung results from a coupling of the initial electron to the electromagnetic field of the nucleus, somewhat screened by the electrons (log term), and augmented by a direct contribution from the electrons (Z2 replaced by Z (Z + 1)). The radiation length of a compound, or mixture, can be calculated using: 1/X0 = Σ wj /Xj

(6.6)

where the wj are the fractions by weight of the nuclear species j of the mixture or of the compound. The spectrum of photons with energy k radiated by an electron of energy E traversing a thin slab of material (expressed as a function of y = k/E) has the characteristic “bremsstrahlung” spectrum:   dσ/dk = A/ (X0 NA k) · 4/3–4/3y + y2 .

(6.7)

At very high energies a number of effects, considered at the end of this subsection, modify the spectrum. Another important quantity, the critical energy can be introduced examining Fig. 6.2. The critical energy Ec for electrons (or positrons) in a given medium is defined as the energy at which energy loss by radiation in a thin slab equals the energy loss by ionization. A slightly different definition ε0 , introduced by Rossi, results from considering the relative energy loss as fully independent of energy (see Fig. 6.2). The critical energy ε0 is well described in dense materials (see Fig. 6.4) by: ε0 = 610 MeV/ (Z + 1.24) .

Fig. 6.4 Critical energy for the chemical elements, using Rossi’s definition [6]. The fits shown are for solids and liquids (solid line) and gases (dashed line)

(6.8)

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Fig. 6.5 Electron-positron pair creation in the field of a nucleus (A, Z)

As will be seen below, X0 and Ec (or ε0 ) are among the important parameters characterizing the formation of electromagnetic showers. Several processes contribute to the interaction of photons with matter, the relative importance of which depends primarily on their energy. Pair Production This process is dominant as soon as photon energies are above a few times 2 me c2 . The graph responsible of the process (Fig. 6.5) shares the vertices of the bremsstrahlung graph. The dominant part (Z2 ) is due to the nucleus, while the electrons contribute proportionally to Z. The process of pair production has been studied in detail [7]. The pair production cross section can be written, in the complete screening limit at high energy as: dσ/dx = A/(X0 NA ) · (1–4/3x (1 − x)) ,

(6.9)

where x = E/k is the fraction of the photon energy k taken by the electron of the pair. Integrating the cross section over E gives the pair production cross section: σ = 7/9 A/(X0 NA ) .

(6.10)

After 9/7 of an X0 , the probability that a high-energy photon survives without having materialized into an electron-positron pair is 1/e. In the pair production process the energy of the recoil nucleus is small, typically of the order of me c2 , implying that at high photon energy (k >> me c2 ) the electron and the positron are both collinear with the incident photon. When the reaction takes place with an electron, the momentum transfer can be much higher leading to “triplets” with one positron and two electrons in the final state. As for bremsstrahlung the cross section is affected at very high energy by processes considered later. Compton Effect The QED cross-section for the photon-electron scattering (Klein-Nishina [8]) can be written in the limit of k >> me c2 , using x = k/me c2 ,   σ = πre 2 (log 2x + 1/2)/x cm2 .

(6.11a)

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The related probability for Compton scattering after the traversal of a material slab of thickness dt and mass per unit volume ρ is: φ = σρ NA Z/A dt.

(6.11b)

For high Z (e.g. lead) the maximum of the Compton cross section and the pair production cross-section are of the same order of magnitude, while for lighter materials the maximum of the Compton cross section is higher. This is illustrated in Fig. 6.6 (from [1]) where carbon and lead are compared. The differential Compton cross-section, with θ denoting the scattering angle between the initial and final photon, and η the angle between the vector perpenFig. 6.6 Photon total cross section as a function of the photon energy in carbon and lead, with the contributions of different processes. σ p.e. corresponds to the atomic photoelectric effect and κ nuc (κ e ) corresponds to pair production in the nuclear (electron) field

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dicular to the scattering plane and the polarization vector of the initial photon (in case it is linearly polarized) reads (ε being the ratio between the scattered and the incident photon energy ε = 1/(1 + k/me c2 (1 − cosθ )): dσ/d = 0.5 re2

  ε + 1/ε − 2 sin2 θ cos2 η .

(6.12)

At low energy (k not larger than a few MeV), the η-dependence can be exploited for polarization measurements (Compton polarimetry). In the same energy range the probability of backward scattering is also sizeable. Photoelectric Effect For sufficiently low photon energies the atomic electrons can no longer be considered as free. The cross section for photon absorption, followed by electron emission (photoelectric effect) presents discontinuities whenever the photon energy crosses the electron binding energy of a deeper shell. Explicit calculations [4] show that above the K-shell the cross section decreases like E−3.5 . In the section devoted to shower formation, the relevance of the photoelectric effect will be considered. The coherent scattering (or Rayleigh scattering) is comparatively smaller than the photoelectric effect and its role negligible for shower formation. High Energy Effects (LPM) In the collinear approximation of bremsstrahlung, the longitudinal momentum difference q|| between the initial electron (energy E) and the sum of the final electron and photon (energy k) is equal to q|| = me 2 c3 k/2E (E − k) .

(6.13)

This quantity can be extremely small, being for example 0.002 eV/c for a 25 GeV electron radiating a 10 MeV photon. Such a small longitudinal momentum transfer implies a large formation length, Lf (Lf q|| ≥ h/2π), about 100 μm in the above example. Secondary interactions (like multiple scattering) taking place over this distance will perturb the final state and will in general diminish the bremsstrahlung cross section and the pair production cross section in case of photon interactions. Coherent interaction of the produced photons with the medium (dielectric effect) also affects, and reduces, the bremsstrahlung cross-section. Such effects, already anticipated by Landau and Pomeranchuk [9] were considered in detail by several authors, and were measured by the experiment E146 at SLAC. A recent overview is given in [10]. The high k/E part of the bremsstrahlung spectrum is comparatively less affected (because of much larger q|| values) while the low k/E part is significantly influenced for E above ~100 GeV, see Fig. 6.7. Only at much higher energies (>10 TeV) is the pair production cross-section affected. In crystalline media the strong intercrystalline electrical fields may result in coherent suppression or enhancement of bremsstrahlung. Net effects depend on the

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10 GeV

(X0NA/A)ydσLPM/dy

1.2

0.8

Bremsstrahlung

100 GeV 1 TeV 10TeV 100TeV

0.4

1 PeV 10 PeV 0

0

0.25

0.5

0.75

1

y=k/E Fig. 6.7 Normalized Bremsstrahlung cross-section k dσ /dk in lead as a function of the fraction of momentum taken by the radiated photon

propagation direction of the particle with respect to the principal axes of the crystal [11]. Hadronic Interactions of Photons Photons with energies above a few GeV can behave similarly to Vector Mesons (ρ, ω and φ) with the same quantum numbers and in this way develop strong interactions with hadronic matter. They can be parameterized with the Vector Meson Dominance model. Using the Current-Field Identity [12], the amplitude for interactions of virtual photons γ∗ of transverse momentum q is:     A (γ∗ A → B) = e/2γρ m2 ρ / m2 ρ − q 2 A (ρA → B) +equiv.terms for ω and φ mesons.

(6.14)

Various photo- and electro-production cross sections were calculated and confronted with experiment. As an example the ratio of hadron production to electronpositron pair production in the interaction of a 20 GeV photon is about 1/200 for hydrogen and 1/2500 for lead [13]. While this ratio is small, the effect on shower characteristics and on particle identification can in certain cases be significant [for example—see Ref. 14—when studying CP violating ππ final states in KL decays, for which πeν decays are a background source].

6.2.2 Electromagnetic Showers When a high energy electron, positron or photon impinges on a thick absorber, it initiates an electromagnetic cascade as pair production, bremsstrahlung and Compton effects generate electrons/positrons and photons of lower energy. Electron/positron energies eventually fall below the critical energy, and then dissipate their energy

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by ionization and excitation rather than by particle production. Photons propagate somewhat deeper into the material, being ultimately absorbed primarily via the photoelectric process. Given the large number of particles (electrons, positrons, photons) present in a high energy electromagnetic cascade (more than one thousand for a 10 GeV electron or photon in lead), global variables have been sought to describe the average shower behaviour. Scale variables, such as X0 as unit length, can be used to parameterize the radiation effects. However, since energy losses by dE/dx and by radiation depend in a different way on material characteristics, one should not expect perfect ‘scaling’. Analytical Description In an analytical description [6] a first simplification consists in ‘factorizing’ the longitudinal development and the lateral spread of showers, with the assumption that the lateral excursion of electrons and photons around the direction of the initial particle does not affect the longitudinal behaviour and in particular the ‘total track length’ (see below). As for any statistical process the first goal is to obtain analytical expressions for average quantities. Particularly relevant (for a shower of initial energy E0 ) are: c(E0 ,E,t) the average number of electrons plus positrons with energy between E and E + dE at depth t (expressed in radiation length), and the integral distribution 0E C(E0 ,E,t) = 0 c(E0 ,E’,t)dE’; n(E0 ,E,t) and N(E0 ,E,t) are the corresponding functions for photons. Using the probability distribution functions of the physical effects driving the shower evolution (Bremsstrahlung, Compton, dE/dx, pair production) one can write and solve [15, 16] ‘evolution equations’ correlating C(E0 ,E,t) and N(E0 ,E,t). In the so called ‘approximation B’ of Rossi, the energy loss of electrons by dE/dx is taken as constant, and the pair production and bremsstrahlung cross-sections are approximated by their asymptotic expression. As an illustration, Fig. 6.8 shows the number of electrons and positrons as a function of depth, in a shower initiated by an electron of energy E0 , and by a photon of energy E0 in units of the “Rossi critical energy ε0 ” (see Sect. 6.2.1). These distributions are integrated over E from 0 to the maximum possible. The area under the curves is to a good approximation equal to E0 /ε0 , in accordance with the physical meaning of ε0 . The two sets of curves also show that a photon initiated shower is shifted on average by about 1 X0 to larger depths compared to an electron (or positron) initiated one. 0∞ The total track length T T L = 0 C(E0 , 0, t)dt the energy transferred to the calorimeter medium by dE/dx, the source of the calorimeter signal. Results from Monte Carlo Simulations While analytical descriptions are useful guidelines, many applications require the use of Monte-Carlo (MC) simulations reproducing step by step, in a statistical manner, the physical effects governing the shower formation. For several decades, the standard simulation code for electromagnetic cascades has been EGS4 [17]. A recent alternative is encoded in the Geant4 framework [18].

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Fig. 6.8 Number of charged secondaries as a function of shower depth, for an electron initiated shower (full lines) and a photon initiated one (dashed lines), calculated analytically by Snyder [15, 16]. The numbers attached to each set of curves indicate ln(E0 /ε0 )

As an illustration of the additional information obtained by this MC approach, Fig. 6.9 shows results of a 30 GeV electron shower simulation in iron (Ec = 22 MeV). The energy deposition per slab (dt = 0.5X0) is shown as a histogram, with the fitted analytical function (see below) superimposed. This distribution is close, but not identical, to the distribution of electrons above a certain threshold (here taken as 1.5 MeV) crossing successive planes (right-hand scale): the energy deposition is slightly below the number of electrons at the beginning of the shower, and somewhat higher at the end. Multiple scattering (see below), affecting more the low energy shower tail, is one effect contributing to this discrepancy. The distribution of photons above the same threshold of 1.5 MeV is shifted to larger X0 with respect to the electron distribution, reflecting the higher penetration power of photons already mentioned. As a further illustration of the power of MC simulations, Fig. 6.10 displays longitudinal profiles of 10 GeV electron showers obtained by Geant4 simulation in lead, copper and aluminium. Since the dE/dx per X0 is relatively more important in low Z material compared to high Z materials, one expects showers to penetrate more deeply in high Z materials, a fact born out by the simulations. Illustrating the energy dependence of shower parameters Fig. 6.11 displays shower energy deposition as a function of depth (shower profiles) for a range of incident electron energies (1 GeV to 1 TeV) in lead. The position of the shower maximum shows

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Fig. 6.9 EGS4 simulation of a 30 GeV electron-induced cascade in iron. The histogram is the fractional energy deposition per radiation length, and the curve is a gamma function fit to the distribution. The full (open) points represent the number of electrons (photons) with energy greater than 1.5 MeV crossing planes at X0 /2 intervals

Fig. 6.10 Fractional energy deposition per longitudinal slice of 1 X0 for 10 GeV electrons in aluminium (full line), copper (dashed) and lead (dash-dotted) (Geant4)

the expected logarithmic dependence on incident energy. In the parameterisation of shower profiles by Longo and Sestili [19]. F(E, t) = E0 b(bt)a−1 e−bt /Γ (a)

(6.15)

one finds accordingly tmax = (a − 1)/b, well fitted by tmax = ln(y) + Ci , (Ci = 0.5 for photons, −0.5 for electrons, and y = E/Ec ).

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Fig. 6.11 Fractional energy deposition in lead, per longitudinal slice of 1 X0 , for electron induced showers of 1 GeV (full line), 10 GeV (dashed), 100 GeV (dash-dotted) and 1 TeV (dotted) (Geant4)

Finally, Fig. 6.12 illustrates the imbalance between electrons and positrons: in an electromagnetic shower, and rather material independent, about 75% of the energy deposited by charged particles is due to electrons, and 25% to positrons. This imbalance is due to the Compton and photoelectric effects which generate only electrons. It is more important towards the end of the shower. Lateral Shower Development Bremsstrahlung and pair creation on nuclei take place without appreciable momentum transfer to the (heavy) nuclei. Bremsstrahlung on electrons of the medium and Compton scattering involve however some momentum transfer. For example, in the Compton interaction of a 2 MeV (0.5 MeV) photon, 6% (16%) of the scattered photons are emitted with an angle larger than 90◦ with respect to the initial photon direction z. Another important effect contributing to the transverse spread in a cascade is multiple scattering of electrons and positrons. After a displacement of length l along z, in a medium of radiation length X0 , the projected rms angular deviation along the transverse directions x and y, of an electron of momentum p is: Es 1 1 l/X0 θx,y = √ 2 pβc

(6.16)

θx,y l δx,y = √ 3

(6.17)

and the lateral displacement is

√ with Es = me c2 (4π/α) = 21.2 MeV. The lateral displacement contributes directly to the transverse shower broadening. If, after a step of length l, the electron emits

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Energydeposit[MeV/X0]

100

e+ e– 10

0

3

6

9

12

15

Depth [X0] Fig. 6.12 Energy deposited in longitudinal slices of 1 X0 by electrons (open symbols) and positrons (closed symbols) in a 10 GeV electron shower developing in lead (EGS4)

a bremsstrahlung photon, the emission will take place along the direction of the electron after l, thus at some angle (rms θ x,y in both directions) with respect to the initial electron. Since the photon travels on average a considerable distance before materializing (9/7 X0 if the photon is above a few MeV, significantly more at lower energy, see Fig. 6.6), the angular deviation of the electron gives a second, large contribution to the shower broadening. In order to quantify the transverse shower spread, it is customary to use as parameter the Molière radius defined as: ρM = Es X0 /Ec ,

(6.18)

√ where ρ M equals 6 times the transverse displacement of an electron of energy Ec , after a path (without radiation nor energy loss) of 1 X0 . The most relevant physical meaning of ρ M comes from Monte-Carlo simulations which show that about 87% (96%) of the energy deposited by electrons/positrons in a shower is contained in a cylinder of radius 1 (2) ρ M . Going back to the expressions of X0 and Ec , it can be seen that their ratio is proportional to A/Z, and thus ρ M is rather independent from the nuclear species, and is essentially governed by the material density. Calculations of ρ M , for some pure materials and mixtures are reported in Table 6.1.

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Table 6.1 Properties of calorimeter materials Material C Al Fe Cu Sn W Pb 238 U Concrete Glass Marble Si Ge Ar (liquid) Kr (liquid) Polystyrene Plexiglas Quartz Lead-glass Air 20◦ , 1 atm Water

Z 6 13 26 29 50 74 82 92 – – – 14 32 18 36 – – – – – –

Density [g cm−3 ] 2.27 2.70 7.87 8.96 7.31 19.3 11.3 18.95 2.5 2.23 2.93 2.33 5.32 1.40 2.41 1.032 1.18 2.32 4.06 0.0012 1.00

Ec [MeV] 83 43 22 20 12 8.0 7.4 6.8 55 51 56 41 17 37 18 94 86 51 15 87 83

X0 [mm] 188 89 17.6 14.3 12.1 3.5 5.6 3.2 107 127 96 93.6 23 140 47 424 344 117 25.1 304 m 361

ρ M [mm] 48 44 16.9 15.2 21.6 9.3 16.0 10.0 41 53 36 48 29 80 55 96 85 49 35 74 m 92

λint [mm] 381 390 168 151 223 96 170 105 400 438 362 455 264 837 607 795 708 428 330 747 m 849

(dE/dx)mip [MeV cm−1 ] 3.95 4.36 11.4 12.6 9.24 22.1 12.7 20.5 4.28 3.78 4.77 3.88 7.29 2.13 3.23 2.00 2.28 3.94 5.45 0.0022 1.99

mip minimum-ionizing particle

Comparing as an illustration lead and copper, one observes that the transverse dimensions of showers expressed in mm are essentially the same (because the transverse profiles are almost identical expressed in ρ M (Fig. 6.14) and the ρ M ’s are similar), while the shower in copper is (in mm) a factor 2.5 longer (because X0 (copper) = 14.3 mm against 5.6 mm for lead). On the other hand, despite being much shorter (in mm), the shower in lead contains about 2.5 more electrons (of lower energy in average) than the shower in copper, in the inverse proportion to their respective critical energies (7.4 MeV for lead against 20 MeV for Copper). The lateral spread of showers is on average narrow at the beginning, where the shower content is still dominated by particles of energy much larger than Ec . In the low-energy tail the shower broadens. Monte Carlo simulations allow studying profiles at various depths. This is illustrated in Fig. 6.13 which shows the 90% containment radius as a function of the shower depth and in Fig. 6.14 which shows the radial profile of showers in three different materials. The broader width in the first 2 or 3 X0 can be associated with backscattering (albedo) from the shower, which competes with the narrow core of the shower in its very early part. There is almost

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Fig. 6.13 90% containment radius R90% (full line), in Molière radius ρ M as a function of shower depth, for 100 GeV electron showers developing in lead. For comparison the longitudinal energy deposition is also shown (dashed line, arbitrary scale) (Geant4)

Fig. 6.14 Fractional energy deposition in cylindrical layers of thickness 0.1 ρ M , coaxial with the incident particle direction, for 100 GeV electron-induced showers in aluminium (dotted line), copper (dashed line) and lead (dash-dotted) (Geant4)

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no dependence of shower transverse profiles (integrated over depth) as a function of initial electron energy.

6.2.3 Homogeneous Calorimeters For reasons explained later, large calorimeter systems are often ‘sampling’ calorimeters. These calorimeters are built as a stack of passive layers, in general of high Z material for electromagnetic calorimeters, alternating with layers of a sensitive medium responding to (‘sampling’ the) electrons/positrons of the shower, produced mostly in the passive layers. A homogeneous calorimeter is built only from the sensitive medium. Provided all other conditions are satisfied (full containment of the shower, efficient collection and processing of the signal) homogeneous calorimeters give the best energy resolution, because sampling calorimeters are limited by ‘sampling fluctuations’ (see Sect. 6.2.4). It is instructive to study first the limitations in the “ideal” conditions of homogeneous calorimeters. We first discuss low-energy applications, where the absorption does not involve showering. As an illustration, Fig. 6.15 shows the extremely narrow lines observed [20] when exposing a Germanium (Li-doped) crystal to a γ source of 108m Ag and 110m Ag. The resolution, at the level of one part in a thousand, is far better than obtained with NaI, a frequently used scintillating crystal (see below). Several Fig. 6.15 Pulse height spectra recorded using a sodium iodide scintillator and a Ge (Li) detector. The source is a gamma radiation from the decay of 108m Ag and 110m Ag. Energies of peaks are labelled in keV

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quantitative studies of the energy resolution of high purity Ge crystals, operated at low temperature (77 K) for γ spectroscopy have been made. A rather comprehensive discussion is given in [21]. After subtraction of the electronics noise, the width of the higher energy lines (above 0.5 MeV) is narrower than calculated assuming statistical independence of the created electron-hole pairs (~2.9 eV are needed to create such a pair). The reason for this was first understood by Fano [22]. Fundamentally it is due to the fact that the pairs created are not statistically independent, but are correlated by the constraint that the total energy loss must precisely be equal to the energy of the incident photon (in the limit of a device in which all energy losses lead to a detected signal, in a proportional way, the line width vanishes). Calling σ the rms of the energy ε used to create an electron-hole pair, the √ √ √ actual resolution should be σ /(ε Np), smaller than 1/ Np by a factor F, where F = (σ /ε)2 is the Fano factor. Monte-Carlo simulations [23] reproduce the phenomenon and give F ~ 0.1 for semiconductor devices, in reasonable agreement with measurements [21]. When two energy loss mechanisms compete, e.g. ionization and scintillation, the total energy constrain remains, but with a binomial sharing between the two mechanisms. It is thus expected that summing up the two contributions, assumed to be read out independently, will lead to an improved energy resolution (it should be remembered however that a certain fraction of the energy lost in the medium goes to heat. This was first demonstrated with a liquid argon gridded cell exposed to La ions with an energy of 1.2 GeV/nucleon traversing the cell [24]. In this set-up both scintillation photons and electrons from electron-ion pairs were detected (see Sect. 6.3.3 for the collection mechanism). More recently, detailed studies of scintillation and ionization yields were made in liquid xenon using 662-keV γ-rays from a 137 Cs source [25]. With decreasing voltage applied to the sensitive liquid Xe volume, the scintillation signal increases while the ionization one decreases, as expected from recombination of electrons-ions giving rise to additional photons. The spectra obtained with scintillation alone, ionization alone, and their sum are shown in Fig. 6.16, together with the correlation between the two signals. The ratio between scintillation and ionization depends also on the nature and energy of the particle making the deposit. Low energy nuclear recoils are highly ionizing, giving rise to more recombination and thus an increased light over charge ratio. As discussed in Sect. 6.3.1, noble liquid detectors (using either argon or xenon) have been developed in the last decade which allowed pushing the limits of dark matter searches. They rely heavily on the existence of two correlated signals (ionization and scintillation) for a given energy deposit, exploiting in particular the ratio between the two to distinguish nuclear recoils from photon or muon background (see Sect. 6.7.2). When the energy loss per unit length becomes very high (i.e. for low values of β and/or high values of the electric charge Ze for ions) saturation effects are observed in liquid ionization detectors, and also in scintillators. Empirically, the

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Fig. 6.16 Correlation between scintillation and ionization signals [25]. Scintillation alone (topleft), ionization alone (top-right), sum of both (bottom-left), 2-D correlation between scintillation and ionization (bottom-right)

effective scintillation (ionization) signal dL/dx (dI/dx) can be parameterized with “Birks law” [26]: dL/dx = L0 .dE/dx/(1. + kB dE/dx),

(6.19)

in which L0 is the luminescence at low specific ionization density. The effect in plastic scintillators, for which kB ~0.01 g.cm−2 MeV−1 , results in suppression (“quenching”) of the light emission by the high density of ionized and excited molecules. Deviations from Birk’s law have been observed for high Z ions [27]. In liquid ionization detectors the effect is associated with electron-ion recombination. It depends upon the electric field, in magnitude and direction with respect to the ionizing track. A typical value in liquid argon is kB ~0.04 g.cm−2 MeV−1 for an electric field in a direction perpendicular to the track of 1 kV/cm, with kB being inversely proportional to E, for E < 1 kV/cm [28].

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Saturation effects are not relevant for electron or photon induced showers (at least below few TeVs) because the track density remains comparatively low (however, depending on the technique used for sampling calorimeters, internal amplification— like in calorimeters with gaseous readout—may saturate for high track density). Saturation effects do affect hadronic showers because of slow, highly ionizing fragments from nuclear break-up and slow proton recoils. The—in general excellent—energy resolution of homogeneous calorimeters used for electromagnetic showers is affected by several instrumental effects. One of the most fundamental ones, the existence of a threshold energy Eth above which an electron of the shower does produce a signal will be illustrated in Sect. 6.3.2 when dealing with Cherenkov based electromagnetic calorimeters. Other effects include: – – – – –

longitudinal and transverse shower containment efficiency of light collection photoelectron statistics electron carrier attachment (impurities) space charge effects, . . .

These effects will be considered when dealing with examples where they are particularly relevant. The closer a detector approaches the intrinsic resolution— like for Ge crystals—the more important are the above limitations. In practice, large calorimeter systems for high energy showers based on homogeneous semiconductor crystals are unaffordable. Scintillating crystals and pure noble liquids are the best compromise between performance and cost, but do suffer from other limitations, as illustrated in examples given below.

6.2.4 Sampling Calorimeters and Sampling Fluctuations In the simplest geometry, a sampling calorimeter consists of plates of dense, passive material alternating with layers of sensitive material. For electromagnetic showers, passive materials with low critical energy (thus high Z) are used, thus maximizing the number of electrons and positrons in a shower to be sampled by the active layers. In practice, lead is most frequently used. Uranium has also been used to optimize the response towards hadrons (Sect. 6.2.7), and tungsten has been used in cases where compactness is a premium. The thickness t of the passive layers (in units of X0 ) determines the sampling frequency, i.e. the number of times a high energy electron or photon shower is ‘sampled’. Intuitively, the thinner the passive layer (i.e. the higher the sampling frequency), the better the resolution should be. The thickness u of the active layer is usually characterized by the sampling fraction fS which is the ratio of dE/dx of a

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minimum ionizing particle in the active layer to the sum of dE/dx in the active and passive layers:    fS = u dE/dxactive / u dE/dxactive + t dE/dxpassive u, t in g cm−2 , dE/dx in MeV/g cm−2 .

(6.20) This ‘sampling’ of the energy results in a loss of information and hence in additional ‘sampling fluctuations’. An approximation [29, 30] for these fluctuations in electromagnetic calorimeters can be derived using the total track length (TTL) of a shower initiated by an electron or photon of energy E. The signal is approximated by the number Nx of e+ or e− traversing the active signal planes, spaced by a distance (t + u). This number Nx of crossings is Nx = T T L/ (t + u) = E/(ε0 (t + u)) = E/ΔE, ΔE being the energy loss in a unit cell of thickness (t + u). Assuming statistical independence of the crossings, the fluctuations in Nx represent the ‘sampling fluctuations’ σ (E)samp, √ √ σ (E)samp /E = σ (Nx ) /Nx = 1/ Nx = {ΔE (GeV) /E (GeV)} √ √ = 0.032 {ΔE (MeV) /E (GeV)} = a/ E.

(6.21)

The detector dependent constant a is the ‘sampling term’ of the energy resolution (see also below). For illustration, for a lead/scintillator calorimeter with 1.4 mm lead plates, interleaved with 2 mm scintillator planes, ΔE = 2.2 MeV, one estimates a ~ 5% for 1 GeV electromagnetic showers. This represents a lower limit (the experimental value is closer to 7 to 8%), as threshold effects in signal emission and angular spread of electrons around the shower axis worsen the resolution [29]. In addition, a large fraction of the shower particles are produced as e+ e− pairs, reducing the number of statistically independence crossings Nx . The sampling fraction fS has practical consequences, considering the actual signal produced by the calorimeter. If fS is too small, the signal is small and may be affected by electronics noise and possibly other technical limitations due to the chosen readout technique (see below). The dominant part of the calorimeter signal is actually not produced by minimum ionizing particles, but rather by the low-energy electrons and positrons crossing the signal planes. Defining the fractional response fR of a given layer “i” as the ratio of energies lost in the active layer and of the sum of active plus passive layers one has i i i fRi = Eactive /(Eactive + Epassive )

(6.22)

with the constraint that  i (E i active + E i passive ) = E. Experimentally one finds that fR (taking all layers together) is significantly smaller than fS [31]. The ratio fR /fS , usually called ‘e/mip’ for obvious reasons, can

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be as low as 0.6 when the Z of the passive material (lead) is much larger than the Z of the active one (plastic scintillator, liquid argon). This effect, well reproduced by Monte-Carlo simulations, is to some extent due to the “transition effect” between the passive and active material, but also due to the fact that a significant fraction of electrons produced in the high Z passive material by pair production or Compton scattering do not have enough energy to exit this layer and are thus not sampled. This same effect induces a depth dependence of e/mip, which decreases by few percent towards the end of the shower. Taking into account an energy independent contribution from electronics noise b, and a minimum asymptotic value of the relative energy resolution c (constant term, due for example to inhomogeneities in materials, imperfection of calibrations, . . . ) the energy resolution of a sampling calorimeter is in general written as1 √ ΔE/E = a/ E ⊕ b/E ⊕ c

(6.23)

Experimentally it has been observed that the same relation holds also for homogeneous calorimeters, in general with smaller ‘sampling terms’ a, although their origin is not coming from sampling fluctuations, but from other limitations (see Sect. 6.2.3).

6.2.5 Physics of the Hadronic Cascade By analogy with electromagnetic showers, the energy degradation of high-energy hadrons proceeds through an increasing number of (mostly) strong interactions with the calorimeter material. However, the complex hadronic and nuclear processes produce a multitude of effects that determine the performance of practical instruments, making hadronic calorimeters more complicated instruments to optimize and resulting in a significantly worse intrinsic resolution compared to the electromagnetic one. Experimental studies by many groups helped to unravel these effects and permitted the design of high-performance hadron calorimeters. The hadronic interaction produces two classes of secondary processes. First, energetic secondary hadrons are produced with momenta typically a fair fraction of the primary hadron momentum, i.e. at the GeV scale. Second, in hadronic collisions with the material nuclei, a significant part of the primary energy is consumed by nuclear processes such as excitation, nucleon evaporation, spallation, etc., generating particles with energies characteristic of the nuclear MeV scale. The complexity of the physics is illustrated in Fig. 6.17, which shows the energy spectra of the major shower components (weighted by their track length in the shower) averaged over many cascades, induced by 100 GeV protons in lead. These spectra are dominated by electrons, positrons, photons, and neutrons at low energy.

1 In

a formula like (6.23), a ⊕ b means

√ 2 (a + b2 ).

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Fig. 6.17 Particle spectra produced in the hadronic cascade initiated by 100 GeV protons absorbed in lead (left). The energetic component is dominated by pions, whereas the soft spectra are composed of photons and neutrons. The ordinate is in ‘lethargic’ units and represents the particle track length, differential in log E. The integral of each curve gives the relative fluence of the particle [32]. On the right, same figure for 100 GeV electrons in lead, showing the much simpler structure, dominated by electrons and photons (hadrons are down by more than a factor 100)

The structure in the photon spectrum at approximately 8 MeV reflects a (n,γ) reaction and is a fingerprint of nuclear physics; the line at 511 keV results from e+ e− annihilation photons. These low-energy spectra encapsulate all the information relevant to the hadronic energy measurement. Deciphering this message becomes the story of hadronic calorimetry. The energetic component contains protons, neutrons, charged pions and photons from neutral pion decays. Due to the charge independence of hadronic interactions, on average approximately one third of the pions produced will be π0 s, fπ0 ≈ 1/3. These neutral pions will decay to two photons, π0 → γγ, before reinteracting hadronically and will induce an electromagnetic cascade, proceeding along its own laws of electromagnetic interactions (see Sect. 6.2.2). This physics process acts like a ‘one-way diode’, transferring energy from the hadronic part to the electromagnetic component, which will not contribute further to hadronic processes. As the number of energetic hadronic interactions increases with increasing incident energy, so will the fraction of the electromagnetic cascade. This simple picture of the hadronic showering process leads to a power law dependence of the two components [33, 34]; naively, the electromagnetic component is Fem = 1 − (1 − fπ0 )n , n denoting the number of shower generations induced by a particle with energy E. For the hadronic fraction Fh one finds in a more realistic evaluation Fh = (E/E0 )k . The parameter k expresses the energy dependence and is related to the average

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multiplicity m of a collision, with k = ln (1 − fπ0 )/ln m. The parameter E0 denotes the average energy necessary for the production of a pion, approximately E0 ≈ 2 GeV; with the multiplicity m ≈ 6–7 of hadrons produced in a hadronic collision k is ≈ −0.2. Values of Fh are of order 0.5 (0.3) for 100 (1000) GeV showers. As the energy of the incident hadron increases, it is doomed to dissipate its energy in a flash of photons. Were one to extrapolate this power law to the highest particles energies detected calorimetrically, E ≤ 1020 eV more than 98% of the hadronic energy would be converted to electromagnetic energy! The low-energy nuclear part of the hadronic cascade has very different properties, but carries the dominant part of the energy in the hadronic sector. In the energetic hadron collisions with the nuclei of the calorimeter material, their nucleons will be struck initiating an ‘intra-nuclear’ cascade. In the subsequent steps, the intermediate nucleus will de-excite, in general through a spallation reaction, evaporating a considerable number of nucleons, accompanied by few MeV γ-emission. The binding energy of these nucleons released in these collisions is taken from the energy of the incident hadron. The number of these low-energy neutrons is large: ~ 20 neutron/GeV in lead. The fraction of the total associated binding energy depends on the incident energy and may be as high as ~20–40%. These neutrons will ultimately be captured by the target nuclei, resulting in delayed nuclear photon emission (at the ~ μs timescale). The energy lost to binding energy is therefore, in general, not detected (‘invisible’) in practical calorimeters. In Fig. 6.18 the energy dependence of the electromagnetic, fast hadron and nuclear components is shown. The response of a calorimeter is determined by the sum of the responses to these different components which react with the passive and Fig. 6.18 Characteristic components of proton-initiated cascades in lead. With increasing energy the em component increases [32]

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active parts of the calorimeter in their specific ways (see Sect. 6.2.7). Contributions from neutrons and photons from nuclear reactions, which have consequences for the performance of these instruments, are also shown in Fig. 6.18. The total energy carried by photons from nuclear reactions is substantial: only a fraction, however, will be recorded in practical instruments, as most of these photons are emitted with a considerable time delay (~1 μs). The event-by-event fluctuations in the invisible energy dominate the fluctuations in the detector signal, and hence the energy resolution. The road to high-performance hadronic calorimetry has been opened by understanding how to compensate for these invisible energy fluctuations [35].

6.2.6 Hadronic Shower Profile The total cross section for hadrons is only weakly energy dependent in the range of few to several hundred GeV, relevant for calorimetry. For protons, the total pp. cross section σ tot is approximately 39 mb. For pion-proton collisions σ tot (πp) = 2/3 σ tot (pp) is naively expected, i.e. 26 mb, compared to the measured value of σ tot (π+ p) ≈ 23 mb. For hadronic calorimetry the inelastic cross sections, σ inel (pA) or σ inel (πA), determine the value of the corresponding interaction length, λint = A/NA σ inel (hadron, A). On geometrical grounds σ inel (hadron, A) is expected to scale as A2/3σ inel (hadron, p), close to the measured approximate scaling A0.71σ inel (hadron, p) and therefore λint ≈ A0.29/{NA σ inel (hadron, p)} [g cm−2 ]. This characteristic length λint is the mean free path of high energy hadrons between hadronic collisions and sets the scale for the longitudinal hadronic shower profile. The probability P(z) for a hadron traversing a distance z without undergoing an interaction is therefore P(z) = exp. (−z/λint ). The equivalence with the characteristic distance X0 for the electromagnetic cascade is evident. In analogy to the parameterization of electromagnetic showers the longitudinal profile of hadronic showers can be parameterized in the form. / . dE/dx = c w{x/X0 }α−1 exp (−bx/X0) + (1 − w) (x/λ)α−1 exp (−dx/λ) . (6.24) The overall normalization is given by c; α, b, d, w are free parameters and x denotes the distance from the shower origin [36]. Longitudinal pion-induced shower profiles are shown in Fig. 6.19 for different energies together with the analytical shower fits. The longitudinal energy deposit rises to a maximum, followed by a slow decrease due to the predominantly lowenergy, neutron-rich part of the cascade. Proton-induced showers show a slightly different longitudinal shape due to the differences in the first few initial collisions. Shower profiles in different materials, when expressed as a function of λint exhibit approximate scaling in λint , in analogy to approximate scaling of electromagnetic

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Fig. 6.20 Longitudinal shower development induced by hadrons in different materials, showing approximate scaling in λ. The shower distributions are measured with respect to the face of the calorimeter (left ordinate). The transverse distributions as a function of shower depth show scaling in λ for the narrow core. The 90% containment radius is much larger and does not scale with λ (right ordinate) [30]

showers in X0 , see Fig. 6.20. Also shown are the transverse shower distributions: the relatively narrow core is dominated by the high-energy (mostly electromagnetic) component. The tails in the radial distributions are due to the soft, neutron-rich, component. In Fig. 6.21 the fractional containment as a function of energy is shown, exhibiting approximately the expected logarithmic energy dependence for a given containment [38, 39].

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Total thickness (λ)

Fig. 6.21 Measured average fractional containment in iron of infinite transverse dimension as a function of thickness and various pion energies [38, 39] 16 MC FCC mean MC FCC peak

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These results indicate that for 98% containment at the 100 GeV scale a calorimeter depth of 9 λint is required. At the LHC, where single particles energies in the multi-hundred GeV and jets in the multi-TeV range have to be well measured, the hadrons are typically measured in 10 λint . For the next jump in collider energy, as is presently studied e.g. for “Future Circular Collider, FCC”, particle and jet energies are approximately a factor 10 higher. For adequate containment, i.e. at the 98% level, calorimeter systems with ~12 λint will be required, see Fig. 6.22 [40]

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6.2.7 Energy Resolution of Hadron Calorimeters The average properties of the hadronic cascade are a reflection of the intrinsic eventby-event fluctuations which determine the energy resolution. Most importantly, fluctuations in the hadronic component are correlated with the number of spallation neutrons and (delayed) nuclear photons and hence with the energy consumed to overcome the binding energy; these particles from the nuclear reactions will contribute differently (in general less) to the measurable signal. Let ηe be the efficiency for observing a signal Ee vis (visible energy) from an electromagnetic shower, i.e., Ee vis = ηe E(em); let ηh be the corresponding efficiency for purely hadronic energy to give a measurable signal in an instrument. Decomposing a hadron-induced shower into the em fraction Fem and a purely hadronic part Fh the measured, ‘visible’ energy Eπ vis for a pion-induced shower is. E π vis = ηe Fem E + ηh Fh E = ηe(Fem + ηh /ηe Fh ) E,

(6.25)

where E is the incident pion energy. The ratio of observable signals induced by electromagnetic and hadronic showers, usually denoted ‘e/π’, is therefore E π vis/E e vis = (e/π)−1 = Fem + ηh /ηe Fh = 1 + (ηh /ηe − 1) Fh .

(6.26)

In general ηe = ηh : in this case, the average response of a hadron calorimeter as a function of energy will not be linear because Fh decreases with incident energy. More subtly, for ηh = ηe , event-by-event fluctuations in the Fh and Fem components produce event-by-event signal fluctuations and impact the energy resolution of such instruments. The relative response ‘e/π ‘turns out to be the most important yardstick for gauging the performance of a hadronic calorimeter. A convenient (albeit non-trivial) reference scale for the calorimeter response is the signal from minimum-ionizing particles (mip) which in practice might be an energetic through going muon, rescaled to the energy loss of a mip. Let e/mip be the signal produced by an electron relative to a mip. Assume the case of a mip depositing e.g. α GeV in a given calorimeter. If an electron depositing β GeV produces a signal β/α, the instrument is characterized by a ratio e/mip = 1. Similarly, the relative response to the purely hadronic component of the hadron shower is ηh Fh E/mip, or h/mip which can be decomposed into h/mip = (fion ion/mip + fn n/mip + fγ γ /mip), with fion , fn , fγ denoting the average fractions of ionizing particles, neutrons and nuclear photons. Practical hadron calorimeters are usually built as sampling devices; the energy sampled in the active layers, fS (Eq. 6.20), is typically a small fraction, a few percent or less, of the total incident energy. The energetic hadrons lose relatively little energy (≤10%) through ionization before being degraded to such low energies that nuclear processes dominate. Therefore, the response of the calorimeter will be

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Fig. 6.23 Conceptual response of a calorimeter to electrons and hadrons. The√curves are for a ‘typical’ sampling calorimeter √ with electromagnetic resolution of σ /E = 0.1/ E(GeV), with hadronic resolution of σ /E = 0.5/ E(GeV) and e/π = 1.4. The hadron-induced cascade fluctuates between almost completely electro-magnetic and almost completely hadronic energy deposit, broadening the response and producing non-Gaussian tails

strongly influenced by the values of n/mip and γ /mip in both the absorber and the readout materials. This simple analysis already provides the following qualitative conclusions for instruments with e/π = 1, as shown conceptually in Fig. 6.23: – fluctuations in Fπ0 are a major contribution to the energy resolution; – the average value (Fem ) increases with energy: such calorimeters have a nonlinear energy response to hadrons; – these fluctuations √ are non-Gaussian and therefore the energy resolution scales weaker than 1/ E. This understanding of the impact of shower fluctuations suggests to ‘tune’ the e/π response of a calorimeter in the quest for achieving e/π = 1, and thus optimizing the performance [41, 42]. It is instructive to analyze n/mip, because of the richness and intricacies of ninduced nuclear reactions and the very large number of neutrons with En < 20 MeV. In addition to elastic scattering a variety of processes take place in high-Z materials such as (n, n’), (n, 2n), (n, 3n), (n, fission). The ultimate fate of neutrons with energies En < 1–2 MeV is dominated by elastic scattering; cross-sections are large (~ barns) and mean free paths short (a few centimetres); the energy loss is ~1/A (target) and hence small. Once thermalized, a neutron will be captured, accompanied by γ-emission. This abundance of neutrons gives a privileged role to hydrogen, which may be present in the readout material. In an n-p scatter, on average, half of the neutron kinetic energy is transferred. The recoil proton, if produced in the active material, contributes directly to the calorimeter signal, i.e., is not sampled like a mip (a 1 MeV proton has a range of ~20 μm in scintillator). The second important n-reaction is the production of excitation photons through the (n,n’,γ) reaction [42].

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This difference in neutron response between high-Z absorbers and hydrogencontaining readout materials has an important consequence. Consider the contributions of n/mip as a function of the sampling fraction fS . The mip signal will be inversely proportional to the thickness of the absorber plates, whereas the signal from proton recoils will not be affected by changing fS : the n/mip signal will increase with decreasing fS . Changing the sampling fraction allows to alter, to ‘tune’ e/π. Tuning of the ratio Rd = passive material [mm]/active material [mm] is a powerful tool for acting on e/π [41]. This approach works well for high-Z absorbers with a relatively large fission cross section, accompanied by multiple neutron emission. Optimized ratios tend to imply for practical scintillator thicknesses rather thick absorbers with concomitant significant sampling fluctuations and reduced signals. How tightly are the various fluctuating contributions to the invisible energy correlated with the average behaviour, as measured by e/π? A quantitative answer needs rather complete shower and signal simulations and confirmation by measurement. Two examples are shown in Fig. 6.24. One observes a significant reduction in the fluctuations and an intrinsic hadronic energy resolution of √ σ /E ≈ 0.2/ E(GeV) for instruments with e/π ≈ 1 [39, 41, 42]. The intrinsic Fig. 6.24 Experimental observation of the consequences of e/π = 1. Shown is the measured pion response in under-compensating, compensating and over-compensating calorimeters; (a) √energy resolution σ /E E as a function of the pion energy, showing deviations from scaling for non-compensating devices. (b) Signal per GeV as a function of pion energy, exhibiting signal non-linearity for non-compensating detectors [41]

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Fig. 6.25 Contributions to and total energy resolution of 10 and 100 GeV hadrons in scintillator calorimeters as a function of thickness of (a) uranium plates and (b) lead plates. The scintillator thickness is 2.5 mm in both cases. The dots in the curves are measured resolution values of actual calorimeters [42]

hadron resolution of a lead-scintillator sampling calorimeter may even be as good √ as σ /E < ≈ 0.13/ E(GeV) [43]. Detectors achieving compensation for the loss of non-detectable (‘invisible’) energy, i.e., e/π = 1, are called ‘compensated’ calorimeters. There are several further negative consequences if e/π = 1 in addition to √ reduced resolution. The energy resolution which no longer scales with 1/ E, is √ usually parameterized as σ /E = a1 / E ⊕ a2 , where a ‘constant’ term a2 is added quadratically, even though physics arguments suggest a2 = a2 (E). Since the fraction of π0 -production Fπ0 increases with energy, such calorimeters have a non-linear energy response. Furthermore, given that the average hadronic fraction Fh are different for pions (Fh (π)) and protons (neutrons) (Fh (p)), typically Fh (π) ~ 0.85Fh(p), the response in calorimeters with e/π = 1 depends on the hadron species [42]. The effects of e/π have been observed [41] (Fig. 6.24) and evaluated quantitatively [42]. Measurements and Monte Carlo simulations of the response of various calorimeter configurations are shown in Figs. 6.25 and 6.26. Besides achieving “intrinsic compensation” with e/π = 1, effective compensation can be achieved by recognizing event by event independently the em fraction Fem and the hadronic fraction Fh , respectively. In instruments with a fine-grained longitudinal and lateral subdivision the different em and hadronic shower shapes provide an approximately independent determination of the two components and the basis for their off-line weighting, resulting in an effective e/π = 1 (see Sect. 6.7.5). Alternatively, the em component and the hadronic component in the shower may be measured independently with a dual readout: one active medium is only sensitive to Cherenkov radiation, predominantly caused by the em component, while the charged particles are measured e.g. with a scintillator, see Sect. 6.3.3. To complete the analysis of the contributions to the energy resolution we need to consider sampling fluctuations, assuming fully contained showers and no degradation due to energy leakage. For electro-magnetic calorimeters a simple expla-

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Fig. 6.26 Monte Carlo simulation of the effects of e/π = 1 on energy resolution (a) and linearity (b) of hadron calorimeters [42]

nation and an empirical parameterization holds (Eq. 6.21): σsamp (em)/E = c(em) · (E(MeV)/E(GeV))1/2, where E is the energy lost in one sampling cell and c(em) ≈ 0.05 to 0.06 for typical absorber and readout combinations. Similar arguments apply for the hadronic cascade; empirically, one has observed [30, 43] that. σsamp (h) /E = c (h) · (ΔE (MeV) /E (GeV))1/2 with c (h) ≈ 0.10.

(6.27)

For high-performance hadron calorimetry sampling fluctuations cannot be neglected. The foundations of modern, optimized hadron calorimetry can be summarized as follows: – the key performance parameter is e/π = 1, which guarantees linearity, E−1/2 scaling of the energy resolution, and best intrinsic resolution; – by proper choice of type and thickness of active and passive materials the response can be tuned to obtain (or approach) e/π ~ 1; – the intrinsic resolution in practical hadron calorimeters can be as good as (σ /E) · √ E < ~ 0.2; – sampling fluctuations contribute at the level of σ /E ≈ 0.10 (E(MeV)/ E(GeV))1/2.

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6.2.8 Muons in a Dense Material The velocity dependence of the average energy loss by collisions of singly charged particles (muons, pions, protons, . . . ) with electrons of the traversed medium differs slightly from formula (6.1) and is given by: 

  Z 1 2me c2 γ2 β2 δ  dE 2 −2 =k − β MeV/ g/cm ln − − dx A β2 I 2

(6.28)

where δ ≈ ln(γ) accounts for screening effects at high energy. As a function of energy of the incident particle the most probable value shows a slow increase (relativistic rise) followed by a plateau whose value depends on the density of the material. The energy loss reaches a minimum for γ β ~ 3, corresponding to muon energies of few hundred MeV. At a given energy, the energy loss distribution of −dE/dx in a slab of material has an asymmetric distribution around its most probable value, usually referred to as the “Landau-Vavilov” distribution [44, 45]. The muon energy loss in dense materials has been extensively studied [46]. Both, the absolute energy loss and the straggling function agree with measurements at the percent level [47] up to several hundred GeV. For muon energies above ~100 GeV, bremsstrahlung, pair production and deep inelastic scattering start to contribute, generating tails in the energy distribution (‘catastrophic energy loss’) [48, 49]. As an illustration, the average contribution of these processes for muons in iron up to 100 TeV is shown in Fig. 6.27. Very roughly speaking a muon behaves as an electron with a critical energy scaled as ≈ (mμ /me )2 . However, unlike for electrons or positrons, pair production is larger than bremsstrahlung. Fig. 6.27 Contributions to the energy loss of muons in iron, as a function of the muon incident energy. The total energy loss in hydrogen gas and uranium is also shown

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Momentum correction to muon momenta can be applied, in setups where muons traverse a calorimeter before entering the muon spectrometer. For muons above ~10 GeV/c there is a good correlation between the total energy loss of muons in a calorimeter with the energy loss recorded in the active medium. This is valuable, particularly for ‘catastrophic’ muon energy loss. Event-byevent correction for the muon energy loss is therefore useful in the hundred GeV momentum range for muon spectrometers behind the calorimeter with few percent momentum resolution [39]. Energy calibration and monitoring is frequently and conveniently done with muons. Exposing a calorimeter to a beam of electrons with well-known energy sets the ‘electron-energy scale’. In sampling calorimeters muons deposing a given energy produce in general more signal than electrons having deposited the same energy: e/μ < 1. While establishing an absolute energy scale with muons requires very careful MC crosschecks, it is very convenient to use muons as a monitor of the calorimeter response as a function of time during data taking and as intercalibration tool between different parts of a calorimeter set-up [50]. The use of muons allows to transfer the absolute energy calibration established in a test beam to the experimental facility and to follow the energy calibration in situ using muons from physics channels. However, given the large dynamic range of energy measurements in many experiments, e.g. at the LHC and the smallness of the muon signal, complimentary calibration methods are necessary to achieve the required accuracy, see Sect. 6.3.6.

6.2.9 Monte Carlo Simulation of Calorimeter Response Modern calorimetry would not have been possible without extensive shower simulations. The first significant use of such techniques aimed to understand electromagnetic calorimeters. For example, electromagnetic codes were used in the optimization of NaI detectors in the pioneering work of Hofstädter, Hughes and collaborators [51]. One code, EGS4, has become the de facto standard for electromagnetic shower simulation [17]. Early hadronic cascade simulations were motivated by experimental work in cosmic-ray physics [52] and sampling calorimetry [53]. However, it were the codes developed by the Oak Ridge group [54], with their extensive modelling of nuclear physics, neutron transport, spallation and fission, which are indissociable from the development of modern hadron calorimetry [35]. Modern, high precision calorimetry and related applications have imposed a new level of stringent quality requirements on simulation: – in many applications, electromagnetic effects have to be understood at the 0.1% level, hadronic effects at the 1% level; – ‘unorthodox’ calorimeter geometries (Sect. 6.7) have to be optimized with simulation tools providing sophisticated interfaces to shower codes:

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– in modern calorimeter facilities the energy deposits are usually distributed over several systems of different geometries and materials. Simulation codes are pushed to their limits in translating the recorded signal into a 1% precision energy measurement; – at LHC and in particular in the study of the UHE Cosmic Ray Frontier simulation codes are used to extrapolate measured detector response by one to eight (!) orders of magnitude; – particle physics MC codes are applied to areas outside particle physics, such as of radiation shielding, nuclear waste incineration and medical radiation treatment. First, we will describe the general approach to these simulation issues before addressing some specific points. Regular conferences on this subject provide a good overview [55]. Electromagnetic Shower Simulation For decades EGS4 [17] has been the standard to simulate electromagnetic phenomena. A modern extended incarnation has been developed by the GEANT4 Collaboration [18]. It includes the full panoply of radiation effects, including photons from scintillation, Cherenkov and Transition radiation up to electromagnetic phenomena relevant at 10 PeV. Hadronic Shower Simulation The simulation must cover the physics and the corresponding cross-sections from thermal energies (neutrons) up to (in principle) the 1020 eV frontier, requiring many different physics models; program suites, ‘toolkits’, such as GEANT4 [18], provide the user with choices of physics interaction models to select the physics interactions and particle types appropriate to a given experimental situation. At high energies (~15 GeV to ~100 TeV)—in addition to measured cross sections—models describing the hadron physics are used, such as the ‘Quark Gluon String’ model [18], Fritiof or Dual Parton Models [56]. Such models are coupled to descriptions of the fragmentation and de-excitation of the damaged nucleus. At the highest energies other models, such as ‘relativistic Quark Molecular Dynamics’ models are being developed [57]. In the intermediate energy range (5

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Fig. 7.27 (a) Ring images from tracks passing through the RICH 1 detector of LHCb, from a single proton-proton collision event at the LHC. (b) Kaon identification efficiency and pion misidentification rate as measured using data. Two different log L (K-π) requirements have been imposed on the samples, resulting in the open and filled marker distributions, respectively. Reference [77]

This approach of peak searching works well in situations of low track multiplicity, where the ring images from tracks are well separated. However, at the LHC the track density is high, as illustrated for a typical event in Fig. 7.27a. In this case the main background to the reconstruction of the ring image of a given track comes from the overlapping rings from other tracks. It is therefore advantageous to consider the optimization of photon assignment to all of the tracks in the event simultaneously, in a so-called global approach. Since a momentum measurement is required to convert a measured ring image into particle identification, as discussed above, it makes sense to use the reconstructed tracks in the event as the starting point for pattern recognition. Trackless ring searches have been developed, but are mostly relevant for background suppression, rather than particle identification [76]. Furthermore, the number of stable charged particle types that are required to be identified is rather limited, typically five: e, μ, π, K, p. The pattern recognition can be made faster by just searching for these particle types, i.e. hypothesis testing. For applications where speed is crucial, such as use in the trigger of the experiment, the number of hypotheses compared can sometimes be further reduced, depending on the physics process that is being selected, e.g. simply comparing π and K hypotheses [79]. On the other hand, if one is interested in an unbiased search for charged particles (such as exotic states) then alternative approaches exist that do not rely on preselected hypotheses [80]. The pattern recognition then proceeds by taking the most likely hypothesis for each of the tracks in the event, typically the π hypothesis as they are the most abundantly produced particle (at the LHC). The likelihood is then calculated that the observed pattern of photons was produced by the particles, under these first choices of mass hypotheses. Conceptually this corresponds to taking the product of terms for each photon according to how close it is to the nearest ring image, assuming

2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0

LHCb

315 Events / (0.02 GeV/c 2)

Events / (9 MeV/c 2)

7 Particle Detectors and Detector Systems

700 LHCb 600 500 400 300 200 100

5

5.1

5.2

5.3

5.4 5.5 5.6 5.7 5.8 π+π– invariant mass (GeV/c 2 )

(a)

0

5

5.1

5.2

5.3

5.4 5.5 5.6 5.7 5.8 π+π– invariant mass (GeV/c 2 )

(b)

Fig. 7.28 (a) Invariant mass distribution for B→ h+ h− decays in the LHCb data before the use of the RICH information, and (b) after applying RICH particle identification. The signal under study is the decay B0 → π + π − , represented by the turquoise dotted line. The contributions from different b-hadron decay modes (B0 → Kπ red dashed-dotted line, B0 → 3-body orange dasheddashed line, Bs →KK yellow line, Bs → Kπ brown line, Λb →pK purple line, Λb → pπ green line), are eliminated by positive identification of pions, kaons and protons and only the signal and two background contributions remain visible in the plot on the right. The gray solid line is the combinatorial background. Reference [81]

a Gaussian probability distribution around each ring. A term is also added to the likelihood from the comparison of the total number of photons assigned to a track, compared to the expected number given the mass hypothesis and momentum. The tracks in the event are then all checked to see which would give the greatest increase in the total likelihood of the event, if its hypothesis were to be changed, and the mass hypothesis of the one giving the greatest increase is then changed. This procedure is iterated until no further improvement in the likelihood can be achieved, at which point the maximum-likelihood solution to the pattern recognition has been found. By the use of various computational tricks [78] this algorithm can be reasonably fast, typically taking a similar CPU time to the track finding algorithm. The performance of this approach to particle identification when applied to LHCb events (of the type shown in Fig. 7.27a) is illustrated in Fig. 7.27b. The efficiency for identifying kaons and the misidentification rate of pions are both shown as a function of momentum. An example from the LHCb experiment of the resulting powerful particle identification in B→ h+ h− decays is shown in Fig. 7.28. The LHCb experiment moves to a fully software trigger where the RICH information is embedded.

7.5 Transition Radiation Detectors A charged particle in uniform motion in free space will not radiate. It can radiate if it traverses a medium where the phase velocity of light is smaller than the velocity of the charged particle. This is Cherenkov radiation as discussed in Sect. 7.4 and was first correctly described by P.A. Cherenkov and S.I. Vavilov in 1934 and formulated

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by I.M Frank and I.E. Tamm in 1937 [15]. This radiation was worked into the BetheBloch formalism in 1940 by E. Fermi, see Chap. 2 and [82]. There is another type of radiation when the charged particle traverses a medium where the dielectric constant, ε, varies. This is transition radiation. It is analogous to bremsstrahlung. In both cases the radiation is related to the phase velocity of the electromagnetic waves in the medium and the velocity of the particle. In the case of transition radiation, the phase velocity changes whereas the particle velocity changes for bremsstrahlung. Transition radiation is, like bremsstrahlung, strongly forward peaked. V.L. Ginzburg and I.M. Frank predicted in 1944 [83] the existence of transition radiation. Although recognised as a milestone in the understanding of quantum mechanics, transition radiation was more of theoretical interest before it became an integral part of particle detection and particle identification [84]. The exact calculation of transition radiation is complex and we will not repeat the mathematics here. The reader is referred to [18, 85, 86]. Specific discussions can be found in [87, 88]. We will here just recall some of the central features. Transition radiation is emitted when a charged particle traverses a medium with discontinuous dielectric constant. Let [E1 , H1 ] be the Lorentz transformed Coulomb field of the charged particle in medium 1 and [E2 , H2 ] the corresponding one in medium 2. See Fig. 7.29a. [E1 , H1 ] and [E2 , H2 ] do not match at the boundary. In order to satisfy the continuity equation, a solution of the homogeneous Maxwell equation must be added in each medium. This is the transition radiation. The angular distribution of transition radiation by a perfectly reflecting metallic surface is of the form: J () = ω

dN α = 2 dωd

π



 −2 γ + 2

2 (7.40)

where γ = E/m  1 in natural units, h¯ = c = 1, α  1/137 is the fine structure constant and   1 is the angle of the photon with respect to the velocity vector v of the charged particle.  is along v for forward transition radiation or its mirror

Boundary

ε2

ε1

Charged particle

Mirror charge

J(Θ) (arbitrary units)

1

0.1

0.01 10 0.1 1 Emission angle Θ (mrad)

(a)

100

(b)

Fig. 7.29 (a) Schematic representation of the production of transition radiation at a boundary. (b) Transition radiation as function of the emission angle for γ = 103 . Eq. (7.40)

7 Particle Detectors and Detector Systems

317

direction for backward transition radiation. N is the total number of emitted photons. Equation (7.40) is plotted in Fig. 7.29b. The energy radiated from a single surface, assuming ε0 → ε, is given by: W =

1 2 αZ ωp γ 3

(7.41)

where ωp is the plasma frequency.

7.5.1 Plasma Frequency The influence of the plasma frequency was shown in the saturation of the relativistic rise expressed by the Bethe-Bloch formula, Chap. 2 and [82], due to the polarisation of the medium: ωp δ 1 = ln + ln βγ − 2 I 2

(7.42)

where I and ωp are respectively the mean excitation energy and the plasma frequency of the medium and δ is the density correction. The plasma frequency, ωp , is the natural frequency of density oscillations of free electrons and its value depends only weakly on the wavelength. Longitudinal plasma waves are resonant at ωp . Transverse electromagnetic waves are absorbed below ωp . If ω < ωp , the index of refraction has an imaginary part and the electromagnetic waves are attenuated or reflected. If ω  ωp , the index is real and a metal becomes transparent. For large ω one can write n2 = 1 −

 ω 2 p

ω

(7.43)

The plasma frequency is given as: ωp2 =

NZe2 ε0 m

(7.44)

and depends only on the total number, NZ, of free electrons per unit volume. The plasma frequency can be approximated with: ωp (eV)  28.8

ρ(g/cm3 ) · z A

(7.45)

where z is the effective number of free electrons per unit volume. Table 7.5 gives the corresponding calculated and measured wavelength, λp , for alkali metals. z = 1 for alkali, group 1a, metals. The calculated plasma energies in Si, Ge and InSb are

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Table 7.5 Ultraviolet transmission limits of alkali metals in nm [89]

Material Li Na K Rb Cs

Table 7.6 Radiator material properties [90]

Material Lithium Beryllium Aluminium Polyethylene Mylar Air

A

Z

λp [nm]

6.939 22.99 39.10 85.47 132.95

3 11 19 37 55

Calculated 155 209 287 322 362

CH2 =CH2 C5 H4 O2

ρ [g/cm3 ] 0.534 1.84 2.70 0.925 1.38 2.2 · 10−3

ωp [eV] 13.8 26.1 32.8 20.9 24.4 0.7

Measured 155 210 315 340 – X0 [cm] 148 34.7 8.91 49 28.7 30.9 · 103

based on four valence electrons per atom. In a dielectric the plasma oscillation is physically the same as in a metal: the entire valence electron sea oscillates back and fourth with respect to the ion core. Table 7.6 tabulates properties of some commonly used radiator material.

7.5.2 Formation Zone A minimum thickness is required in order to efficiently produce the transition radiation as the evanescent field has a certain extension. This is the formation zone and is illustrated in Fig. 7.30 for a stack of aluminium, ωp (Al) ∼ 32.8 eV, and air, ωp (air) ∼ 0.7 eV. The length of the formation zone, d, can be written as:

 ω 2 −1 2c p −2 2 d= γ + + ω ω

(7.46)

√ which has a maximum, dmax , at ω = γ ωp / 2 for  = γ −1 , which is equivalent to the maximum intensity as can be seen from Eq. (7.40) and Fig. 7.29b. dmax (μm) ∼ 140 · 10−3

γ ωp (eV)

(7.47)

Inserting Eq. (7.45) in Eq. (7.47), we see that for media with a density in the order of 1, ωp  20 eV and dmax  7 μm for γ = 1000. For a gas, ωp is about 30 times smaller due to the reduced density and dmax thereby 30 times longer for same γ .

7 Particle Detectors and Detector Systems

319

40

Relative Intensity

4 GeV 30 3 GeV

20

2 GeV 10 1 GeV 0 0

10

20

30

40

50

Air Spacing (mil)

Fig. 7.30 Relative intensity of transition radiation for different air spacing. Each radiator is made of 231 aluminium foils 1 mil thick. (1 mil = 25.4 μm). Particles used are positrons of 1–4 GeV energy (γ = 2000–8000). Adapted from [91]

Using numbers for the experimental set-up in Fig. 7.30, we get dmax ∼ 1.5 mm for γ = 8000.

7.5.3 Transition Radiation Detectors From the discussion above, transition radiation can be characterized by the following: • • • •

Transition radiation is a prompt signal. Transition radiation is not a threshold phenomenon. The total radiated power from a single interface is proportional to γ . The mean emission angle is inversely proportional to γ . In general terms, there are two different types of transition radiation detectors:

1. The detectors working in the low energy, optical, range. 2. The detectors working in the X-ray range. We will briefly introduce the first one and use a little more space on the second class of X-ray transition radiation detectors.

7.5.3.1 Optical Transition Radiation Detectors J.E. Lilienfeld [92] was probably the first,22 in 1919 to observe that in addition to X-rays, radiation ranging from visible light through the ultraviolet is emitted 22

This statement has been contested over the years and could be due to a confusion between transition, Cherenkov radiation and bremsstrahlung. See [93].

320

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SEM Grid

Beam

30o 30o

Intensity monitor

L1

0.6

σ rms (mm)

OTR radiator

650 nm

0.7

450 nm 0.5 0.4 0.3 0.2

L2

0.1

CCD camera

0

500

1000 1500 2000 2500 3000 3500

Intensity (A.U.)

(b)

(a)

Fig. 7.31 (a) Sketch of an experimental set-up for measurement of optical transition radiation with secondary emission, SEM, grid and beam intensity monitor. The transition radiation foil is tilted by 30◦ with respect to the beam line. The optical system is defined by two lenses and a CCD camera. (b) Measured rms beam size values as a function of the total intensity for λ = 450 and 650 nm at 2 GeV. Adapted from [94]

when electrons approach a metallic surface. This radiation has a characteristic polarisation, spectrum and intensity. A variation to this radiation occurs when the charged particle moves roughly parallel to a conducting undulated surface. An oscillating dipole will be set up with a frequency related to the particle velocity and the undulation. The radiated power is small, but due to the microscopic source area, the brightness can be large. This has, amongst a range of other usages, found an application in accurate beam diagnostics equipments. As an example, we will use an experiment to investigate the geometrical resolution of optical transition radiation as shown in Fig. 7.31a [94]. Integrating Eq. (7.40) over the solid angle gives: dN 2α  ln (γ max ) dω πω

(7.48)

where max is the angle of maximum emission, measured by the optical spectrometer. The number of photons emitted is small. This must be compensated by a large number of particles in the beam. The mathematics for such a set-up is given in [95]. The diffraction, or the Heisenberg uncertainty principle in the transverse phase-space of the photon, sets the lower limit for the size of the emitting surface: bi ≥

λ 1 2π 2i

(i = x, y)

(7.49)

where λ ∼ 600 nm is the observed wavelength. bi and i are the components of the impact parameter b and the photon direction. i and bi refer to rms values. Setting  = γ −1 , or full acceptance for the photons, the resolution becomes

7 Particle Detectors and Detector Systems Table 7.7 Parameters for the fit to the data [94] and plotted in Fig. 7.31b

321 Parameter ρ a b

λ = 450 nm 176 ± 12 μm (9 ± 5) · 10−5 1.12 ± 0.09

λ = 650 nm 163 ± 25 μm (6 ± 3) · 10−5 1.12 ± 0.06

proportional to γ . γ = 105 would give b ≥ 5 mm. This effect can be limited by the introduction of an iris in the optical path as in [96]. The results from [94] are shown in Fig. 7.31b. As expected, the resolution is weakly dependent on the intensity of the beam, 1but the total uncertainty is small. The measurement points are fitted to σrms = ρ 2 + aI b , where a and b are fit parameters, ρ is the real beam dimension and I is the beam intensity. These are given in Table 7.7. Another promising application for optical transition radiation is in aerogel23 Cherenkov detectors [97].

7.5.3.2 X-ray Transition Radiation Detectors Following [98], the total radiated energy from a single surface per unit of frequency, can be approximated by:

dW dω

 s.s.

α = π



1 + r + 2X12 X12 + 1 ln 2 −2 1−r X1 + r

 (7.50)

where X1 =

ω γ ωp1

and r =

2 ωp2 2 ωp1

∼

ρ2 ρ1

(7.51)

The suffix 1 and 2 denote medium 1 and 2. ωpi is the plasma frequency for medium i. r will be assumed to be small and in the range of 10−3 , which corresponds to a ρ = 1 to gas interface. By analysing Eq. (7.50), which is plotted in Fig. 7.32a, three distinct regimes can be examined: 1. If γ  ω/ωp1 then X1  1 and dW/dω ∼ α/6πX14 , which is a small number. This results in a frequency cut-off and thereby ω ≤ γ ωp1 . 2. If ω/ωp1  γ  ω/ωp2 then dW/dω ∝ ln X1−1 . That is, the total radiated power increases logarithmically √ with γ . 3. If γ  ω/ωp2 then X1  r. Then the total radiated power is approximately constant.

23 See

Sect. 7.4.2.2.

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0.01

2

dW/dω

dW π single surface dω α

3

1

N foils

0.0001 with

dW π ≈− 1.6 (ln X + 7.0) dω α

0 0.01

0.1

X=

0.001

Single surface

N foils without absorption

absorption

ω γ ⋅ ωp1

1

10

1

10

100 ω (keV)

1000

(b)

(a)

Fig. 7.32 (a) Total radiated energy from a single surface per unit of frequency as function of the dimensionless variable X = ω/γ ωp1 . (b) Intensity of the forward radiation divided by the number of interfaces for 20 μm polypropylene (ωp = 21 eV) and 180 μm helium (ωp = 0.27 eV). Adapted from [99] 10000

1 2 3

k k+1

N

Li t=40 μm

eff

1000

N

γ q

100

Polyethylene t=20 μm

10

Mylar t=15 μm

1

l1

1

l2

(a)

10 ω (keV)

100

(b)

Fig. 7.33 (a) Sketch of a periodic transition radiation radiator. (b) The effective number of foils in a radiator as function of photon energy. Adapted from [90]

It can be shown that the mean radiated energy in this single surface configuration can be written as: W  2αγ ωp1 /3

(7.52)

and that the number of high energy photons produced are of the order of α when taking into account the frequency cut-off discussed above: Nphotons(ω > 0.15γ ωp1)  α/2

(7.53)

A large number of interfaces are therefore required to have an effective detector with a sufficient signal-to-noise ratio. A periodic transition radiation radiator is sketched in Fig. 7.33a. It should be noted that the radiators do not need to be rigorously periodic, but it is helpful for the calculation of the yield.

7 Particle Detectors and Detector Systems

323

The basic mathematics can be found in [85, 90, 98]. Computational models can be found in [100]. The effective final number of transition radiation high energy photons at the end of the radiator stack is a function of constructive and destructive effects. See Fig. 7.32b. We will list the main effects here: • The total radiated energy of a single surface is proportional to the plasma √ frequency and thereby proportional to Z of the material. Equation (7.44). The absorption of these photons is governed by photo-electric effects and the absorption coefficients in the stack. This goes approximately like Z 5 . The radiator material should therefore be of low Z. • The thickness of the radiator material, l1 in Fig. 7.33a, must be large enough to contain the formation zone for the required γ , but short enough not to introduce multiple scattering effects and bremsstrahlung. The gas density will always introduce a negative effect and should be kept as low as possible. For a practical transition radiation radiator and following [90], the expression of the total flux, is then represented by an integration over the emission angle  and a function which represents the incoherent addition of the single foil intensities and includes the photon absorption in the radiator. The effective number of foils in the radiator can then be expressed as: Neff 

1 − exp(−Nσ ) 1 − exp(−σ )

(7.54)

where σ = (κρt)foil + (κρt)gas and κ, ρ and t are respectively the absorption coefficient, density and thickness of the material. The self-absorption of the + photons * from transition radiation limits the yield and Neff → 1/ 1 − exp(−σ ) for N → ∞. A typical mean energy for the photons in a practical radiator is in the range of 10 keV. See Fig. 7.32b. The spectrum will be softer for foils with lower plasma frequencies. Since Neff in Eq. (7.54) is depending on the absorption coefficient through the frequency of the photon, Neff will saturate for high frequencies as shown in Fig. 7.33b.

7.5.3.3 X-ray Detectors Any detector which has a sufficiently high efficiency for X-rays of the order of 10 keV can be used. In the design of the detector it should be noted that the number of transition radiation photons is small and produced very close to the path of the charged particle which will normally also traverse the detector. The traditional detector is a MWPC-like detector, Chap. 3, which directly follows the radiator. In order to enhance the signal-to-noise ratio and efficiently use the space as the effective number of interfaces in the radiator will saturate, a transition radiation detector is therefore normally many radiator/detector assemblies.

R. Forty and O. Ullaland

10000 1000

Ion pairs/cm at NTP

Mass attenuation coefficient (cm 2/g)

324

Xenon Argon

Krypton

100 10 Neon

1

300

Xe

C4 H10

200

Kr CO2

100

O2 Ne

Helium H2

0.1

10

Photon energy ω (keV)

100

Ar N2

He

0 1

CH 4

0

(a)

50

100

A

(b)

Fig. 7.34 (a) X-ray mass attenuation coefficient, μ/ρ, as function of the photon energy. μ/ρ = σtot /uA, where u = 1.660 × 10−24 g is the atomic mass unit, A is the relative atomic mass of the target element and σtot is the total cross section for an interaction by the photon. Data from http:// physics.nist.gov/PhysRefData/. (b) The (×) primary and (+) total number of ion pairs created for a minimum ionizing particle per cm gas at normal temperature and pressure as function of molecular mass A [101]

The ionization loss, dE/dx, from the charged particle will create charge clusters. Some of them rather far from the track due to δ-electrons. The absorption of transition radiation photons will produce a few local strong charge clusters. The choice of gas is therefore a compromise between photon absorption length, Fig. 7.34a, and the background from dE/dx, Fig. 7.34b. The optimal gas thickness is about one absorption length for 10 keV. Xenon is the preferred gas with a chamber thickness of 10–15 mm. See discussion in [90]. CO2 , or similar, is added as quencher. A minimum ionizing particle, MIP, will produce a total of ∼310 ion pairs per cm xenon gas. Figure 7.34b. The relativistic rise is about 75% in xenon at 1 atm, or about 550 ion pairs/cm will be produced by a high γ charged particle. The average energy required to create an ion pair in a gas, is typically 25–35 eV. For xenon it is measured to 22.1 ± 0.1 eV [102], or about double the ionization energy for the least tightly bound shell electron. A 10 keV transition radiation photon will then produce about 450 ion pairs. The signal-to-noise ratio will be further reduced due to Landaufluctuations and gain variations in the detector and electronics. Additional background might arise from curling in a magnetic field, bremsstrahlung and particle conversions. The challenge is then to correctly identify the photon cluster from a dE/dx signal of about the same strength. We will illustrate this by looking more closely at the choices made by the ALICE [103] and ATLAS [104] experiments.

7.5.3.4 ATLAS Transition Radiation Tracker In the ATLAS experiment, the transition radiation tracker (TRT) in the barrel comprises many layers of gaseous straw tube elements interleaved with transition radiation material. Figure 7.35. With an average of 36 hits per track, it provides continuous tracking to enhance the pattern recognition and improve the momentum

7 Particle Detectors and Detector Systems

325 Radiator Sheets

Tension Plate High Voltage Plate Electronics Protection Board Carbon-Fiber Shell Divider

Eyelet Taper Pin

Straws Straw Endplug Wire Support Capacitor Barrel Capacitor Assembly

(a)

(b)

Fig. 7.35 (a) ATLAS Detector. Drawing showing the sensors and structural elements traversed by a charged track of 10 GeV pt in the barrel inner detector (pseudo rapidity η = 0.3). The track traverses approximately 36 axial straws of 4 mm diameter contained in the barrel transitionradiation tracker modules. [104]. (b) Layout of an ATLAS Barrel TRT module. The ATLAS TRT collaboration et al. [105] with permission

resolution over | η |< 2.024 and electron identification complementary to that of the calorimeter over a wide range of energies. A similar detector is placed in the forward direction. The transition radiator material which completely surrounds the straws inside each module, Fig. 7.35b, consists of polypropylene-polyethylene fibre mat about 3 mm thick. The fibres are typically 19 μm in diameter and are formed from polyethylene clad polypropylene material. The fibres are formed into fabric plies with 3 mm thickness and a density of about 0.06 g/cm3 . The absorption length for the lowest energy photons of interest (5 keV) is about 17 mm in the radiator material. The ATLAS TRT uses two thresholds to discriminate between digitisations from tracks and those from transition radiation: 1. Low threshold, LT, for tracking which is set to ∼300 eV with 8 digitisations over 25 ns. 2. High threshold, HT, set in the range 5–7 keV with 1 digitisation over 25 ns and read out in 75 ns segments. As the βγ of the traversing particles will vary greatly, and thereby the ionization in the straw tubes, a Time-over-Threshold parameter can be defined from the LT digitisations in order to enhance the signal-to-noise estimate for the transition radiation signal. Particle identification properties of the TRT Barrel using transition radiation were studied at several different beam energies. The good agreement between 2 GeV low Pseudo rapidity, η, is describing the angle of a particle relative to the beam axis. η =  *  + L − ln tan 2 .  is the angle between the particle momentum and the beam = 12 ln |p|+p |p|−pL axis, p is the momentum vector and pL is the longitudinal momentum component.

24

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Pion efficiency (%) at 90 % of electron

1

Pion efficiency

Test beam Monte Carlo 0.1

0.01

0.001 0.6

0.7

0.8

0.9

1

10 20 GeV, barrel position 9 short barrel full barrel

8 6 4 2 0 3

Electron efficiency

(a)

4

5

6

7

8

9

10

High threshold (keV)

(b)

Fig. 7.36 (a) ATLAS TRT test beam. Pion rejection curve for a 2 GeV e/π beam. Cornelissen and Liebig [106] with permission. (b) ATLAS TRT test beam. e/π rejection power as a function of the high level threshold. Full barrel: all barrel straw layers are active. Short barrel: particle crosses the barrel in the central area where the first 9 layers do not have active anode wires. The ATLAS TRT collaboration et al. [105] with permission

energy data and simulation is shown in Fig. 7.36a. The results for 20 GeV beam energy are shown in Fig. 7.36b. On this figure the pion rejection power is shown as a function of the high level threshold at two beam positions along the straw. The upper points are when beam particles crossed the Barrel module 40 cm from its edge. At this position the first 9 straw layers are not active. The lower points are when the beam is positioned 20 cm from the edge of the Barrel where all 73 straw layers are active. As seen in this figure the best particle identification properties for the TRT Barrel are at a threshold of about 7 keV. Pion mis-identification in that case is 1.5–3% at 90% of the electron efficiency.

7.5.3.5 ALICE Transition Radiation Detector The main purpose of the ALICE Transition Radiation Detector (TRD) [103, 107] is to provide electron identification in the central barrel for momenta above 1 GeV/c. Below this momentum electrons can be identified via specific energy loss measurement in the TPC. Above 1 GeV/c transition radiation from electrons passing a radiator can be exploited together with the specific energy loss in a suitable gas mixture to obtain the necessary pion rejection capability. The chamber geometry and the read-out electronics were chosen to reconstruct track segments. Since the angle of the track segment with respect to the origin is a measure of the transverse momentum of the electron, this information is used in the first level trigger within 5 μs of the collision. The pion rejection is governed by the signal-to-background ratio in the measurement of J/! production and its pt dependence. This led to the design goal for the pion rejection capability of a factor 100 for momenta above 1 GeV/c in central PbPb collisions.

7 Particle Detectors and Detector Systems

TRD stack

327

TRD supermodule

TRD chamber TPC heat shield

TOF

(a)

(b)

Fig. 7.37 (a) Schematic drawing of the TRD layout in the ALICE space frame. Shown are 18 super modules each containing 30 readout chambers (red) arranged in five stacks of six layers. One chamber has been displaced for clarity. On the outside the TRD is surrounded by the TimeOf-Flight (TOF) system (dark blue). On the inside the heat shield (yellow) towards the TPC is shown. The ALICE Collaboration et al. [103] with permission. (b) The principle design of the TRD sandwich radiator. The ALICE Collaboration et al. [107] with permission

The TRD consists of 540 individual readout detector modules. Figure 7.37a. Each detector element consists of a carbon fibre laminated Rohacell25 /polypropylene fibre sandwich radiator, Fig. 7.37b, of 48 mm thickness, a drift section of 30 mm thickness, or about 2 μs, and a multi-wire proportional chamber section (7 mm) with pad readout. Following [108], employing the drift time information in a bidimensional likelihood [109], the pion rejection capability can be improved by about 60% [110] compared to the standard likelihood method on total deposited charge. This method is the simplest way of extending the standard method. However, it does not exploit all recorded information, namely the amplitude of the signal in each time bin. Along a single particle track this information is highly correlated, Fig. 7.38a, due to • the intrinsic detector signal, in particular since a Xe-based mixture is used • the response of the front-end electronics used to amplify the signals. Under these circumstances, the usage of a neural network (NN) algorithm is a natural choice for the analysis of the data. The result of the data analysis from a 2–6 GeV/c mixed e/π test beam is shown in Fig. 7.38b [108]. Neural Network algorithm might improve the pion rejection significantly by a factor larger than 3 for a momentum of 2 GeV/c compared to other methods. The detector was completed in the LS 1 before RUN 2 at LHC. Since then it provides coverage of the full azimuthal acceptance of the central barrel. Figure 7.39 shows the pT spectra of electron candidates with 6 layers identified using the TPC and the TOF in the minimum-bias and triggered data sample. The expected onset

25 ROHACELL is a close cell polymethacrylimide- (PMI-) rigid foam by Evonik Industries AG, Germany.

328

R. Forty and O. Ullaland cathode pads

pion

electron

electron

cathode pads 3

4

5

anode wires

region

anode wires

region

cathode wires

cathode wires

Drift Chamber

Signal

100 75 50 25

drift region

drift region

0 2

bin Time

4

Drift Chamber

primary clusters

6 8 10 12 14

entrance window

1

2

3

4

5

6

7

8

Pad number

x

x

Radiator

Radiator y

z

pion TR photon

electron

electron

(a) 5

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(b) Fig. 7.38 (a) Schematic cross-sectional view of an ALICE detector module in rz and rφdirection. The inset shows the charge deposit from an inclined track which is used for momentum reconstruction. The ALICE Collaboration et al. [103] with permission. (b) Measured pion efficiency as a function of beam momentum applying likelihood on total deposited charge (LQ) (full symbols) measured with a stack of six chambers and smaller test chambers. Results are compared to simulations (open symbols) for 90% electron efficiency and six layers. These simulations were extended to two-dimensional likelihood on deposited charge and position (LQ1, Q2) and neural networks (NN). The ALICE Collaboration et al. [103] with permission

at the trigger threshold of 3 GeV/c is observed for the triggered events and shows in comparison to the corresponding spectrum from minimum-bias collisions an enhancement of about 700. At 90% electron efficiency, a pion rejection factor of

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1 ALICE Pb–p √s = 5.02 TeV NN 1/Nevt dN/dpT (c/GeV)

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offline matching Minimum bias

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Fig. 7.39 pT spectra √ of identified electrons for the minimum-bias and TRD-triggered data sample of Pb-p collisions at s NN = 5.02 TeV. For the result of the TRD-triggered sample, electrons from photon conversions in the detector material were rejected by matching the online track with a track in the TPC. Reference [111]

about 70 is achieved at a momentum of 1 GeV/c for simple identification algorithms. When using the temporal evolution of the signal, a pion rejection factor of up to 410 is obtained.

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Chapter 8

Neutrino Detectors Leslie Camilleri

After a brief introduction describing the many sources of neutrinos, this article will describe the various detector techniques that are being used to observe neutrinos of energies ranging from a few MeV to hundred’s of GeV.

8.1 Historical Introduction In 1930 in order to explain the continuous energy spectrum of electrons emitted in beta decay, Pauli postulated [1] that these electrons were emitted together with a light neutral particle. This particle was subsequently named the neutrino. Their actual observation had to wait until 1953 when Reines and Cowan recorded [2] interactions of anti(electron)neutrinos emitted by a reactor in a cadmium doped liquid scintillator detector. Since then, in addition to the νe , two other flavours of neutrinos were observed, the νμ and ντ . The νμ , which is produced in π → μ decay, was proved to be different [3] from the νe in an experiment at Brookhaven using thick-plate optical spark chambers. The ντ , companion of the τ lepton, was observed [4] at Fermilab in an emulsion cloud chamber detector consisting of iron plates interleaved with sheets of photographic emulsions. Although until recently neutrinos were thought to be massless and were described as such in the Standard Model, in the past decade they have been found to be massive [5, 6]. Furthermore each of the three flavour states mentioned above consists of a superposition of three mass states of unequal masses leading to oscillations of one flavour into another under the appropriate conditions. The characteristics of these oscillations depend on three mixing angles θ13 , θ12 and θ23 as well as on the difference of the square of the 3

L. Camilleri () Nevis Labs, Columbia University, Irvington-on-Hudson, NY, USA e-mail: [email protected] © The Author(s) 2020 C. W. Fabjan, H. Schopper (eds.), Particle Physics Reference Library, https://doi.org/10.1007/978-3-030-35318-6_8

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neutrino masses, m212 referred to as the solar mass difference as it is of importance in oscillations of solar neutrinos, m213 ∼ m223 referred to as the atmospheric mass difference as it drives oscillations of neutrinos produced in the atmosphere through the decay of mesons produced in cosmic ray interactions. The flavour of interacting neutrinos can only be determined if the interaction is via a charged current. In these, the νe , νμ and ντ respectively produce a negative electron, muon or τ lepton in the final state. Antineutrinos produce the corresponding positive charged lepton.

8.2 Sources of Neutrinos and Their Characteristics Naturally occurring neutrinos and man-made neutrinos are produced through several different processes. Nature provides us with solar neutrinos emitted by the sun, atmospheric neutrinos produced by the interaction of cosmic rays in the atmosphere, cosmological neutrinos produced by a variety of deep space violent events, geological neutrinos produced by nuclear decays in the earth core as well as neutrinos produced in beta decay. Man made neutrinos are produced by nuclear reactors or by specially designed beams at accelerators or by highly radioactive sources. These processes are briefly described below.

8.2.1 Solar Neutrinos They are emitted in nuclear reactions occurring in the sun [7]. The three main reactions are p + p → d + e+ + νe , emitting a continuous spectrum of neutrinos with an end point at 0.4 MeV, e+7 Be → 7 Li + νe with a monochromatic spectrum at 0.862 MeV and 8 B →8 Be∗ + e+ + νe also with a continuous spectrum with an end point at 15 MeV. Their total flux on earth is 6.4 × 10+10 cm−2 s−1 .

8.2.2 Atmospheric Neutrinos Atmospheric neutrinos [8] are produced in the decays of π and K mesons produced in the interactions of cosmic rays in the upper atmosphere. Their energy ranges over several orders of magnitude up to hundreds of GeV. They are observed either coming from above or from below and in the latter case they will have traversed the earth. This allows us to observe them from a few kilometers to about 13,000 km from their production point, thus providing us with very different baselines over which to study oscillations. These predominantly νe and νμ neutrinos are usually observed through their charged current interactions respectively producing electrons or muons.

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8.2.3 Cosmological Neutrinos The study of cosmological neutrinos [9] at the TeV scale is in its infancy. Their very low rate necessitates extremely large detectors. This has led to the use of naturally occurring detection media such as lake or sea water and Antarctic ice. The Cerenkov light or radio waves emitted by charged particles produced in their interactions in the medium are recorded, respectively, in strings of photomultiplier tubes or antennas.

8.2.4 Reactor Neutrinos Nuclear reactors are an abundant source of antineutrinos, 6 ν¯ e per nuclear fission on average, resulting in a flux of 1.8 × 1020 per GW thermal energy, emitted isotropically. The standard method to study them [10] is to observe the Inverse Beta Decay (IBD) reaction ν¯ e + p → e+ + n in a hydrogen-rich liquid scintillator detector. In addition to observing photons emitted as a result of the positron annihilation, the neutron can be detected by recording photons emitted by the neutron capture in the scintillator.

8.2.5 Accelerator Neutrinos Accelerator neutrinos are produced [11] by the decay of π and K mesons themselves produced by the interaction of a proton beam on a target as illustrated in Fig. 8.1. The target must be thick enough along the beam to maximize the proton interaction probability and yet thin enough to minimize the reinteraction probability and multiple scattering of the produced mesons such as to produce as high an energy and as focussed a beam as possible. The usual target geometry consists of a series of thin rods of low Z material such as carbon or beryllium separated by a few cms but in line with the proton beam. The mesons are then focussed by a system of toroidal magnets. These, referred to as horns [12], consist of two concentric current sheets, parabolically shaped that provide a toroidal magnetic field. Its strength is inversally proportional to the radial displacement from the beam axis and the integral is such as to bend more the particles that are further away from the beam thus providing a near parallel beam. A second horn is usually provided such as to compensate for

Decay pipe π

Hadron

+

μ+

Absorber Target

Horns

Fig. 8.1 The principle of an accelerator produced neutrino beam

ν Rock

Exp.

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50 GeV 20 GeV 100 GeV

Target Horn Reflector

Fig. 8.2 The principle of horn focusing of mesons in a neutrino beam

over-focussed and under-focussed particles as shown in Fig. 8.2. The particles then enter a long evacuated decay tunnel in which π → μνμ , K → μνμ and K → πeνe decays occur producing a predominantly νμ beam with an admixture of ∼1% of νe . Focussing positive mesons produces a neutrino beam whereas focussing negative mesons, achieved by a reversal of the polarity of the horns, produces an antineutrino beam. An alternative to the horns is a system of bending magnets and quadrupoles. Such a technique [13] has been used in the Sign Selected Quadrupole Train, SSQT, neutrino beam at Fermilab. Its performance is described in [14]. An advantage of this technique is that the neutrino beam not being along the axis of the proton beam, νe from KL◦ decays will not enter the detector since their parents will not be deflected. This is a distinct advantage in oscillation experiments looking for νe appearance in a νμ beam in which the intrinsic νe background is irreducible. The above techniques produce a beam with a broad energy spectrum, referred to as a broad band beam. A narrower range of neutrino energies is sometimes desirable. Such narrow band beams are obtained by first momentum-selecting the parent pions and kaons before they decay using standard beam optics methods, thus reducing the range of neutrino energies. Furthermore, the neutrino energy can be deduced on an event by event basis as it is related to the neutrino production angle and this can be computed from the radial position of the event within the detector. The uncertainty on the energy depends on the momentum and angular spread of the meson beam and on the length of the decay channel. It is typically 5–20%. The intensity of these narrow band beams is necessarily lower than that of broad band beams. Another way to expose the detector to neutrinos with a given narrow energy spectrum is to place the detector at an off-axis angle to the beam [15]. The kinematics of pion decay, shown in Fig. 8.3 are such that neutrinos observed at a non-zero angle to the proton beam have an approximately unique momentum irrespective of the momentum of their parent meson. Furthermore the value of this unique momentum depends on the off-axis angle, thus allowing a detector to be exposed to the neutrino momentum required by the physics under investigation by placing it at the appropriate angle.

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θ = 0 mrad θ = 7 mrad θ = 14 mrad θ = 27 mrad

E [GeV]

15

10

5

0 0

10

20

30

40

E [GeV] Fig. 8.3 The correlation between the pion momentum and its decay neutrino momentum, plotted for several neutrino directions relative to the proton beam axis

An accelerator neutrino beam can also, potentially, contain ντ ’s. These are produced through the production of Ds mesons in the initial proton interactions and their subsequent decays Ds → τ ντ followed by τ → ντ + . . . . However in most accelerator beams the ντ content is negligible since the Ds production cross section is small at existing energies. One notable exception will be discussed in Sect. 8.3.4. The semi-leptonic decays of charmed particles have also been used to produce neutrinos. In this case, because of the very short lifetime of charm, a decay tunnel is unnecessary. The beam is produced in a so-called beam dump [16], in which the incident proton beam and secondary pions and kaons are absorbed before they can decay. At CERN, the beam dump [17] was made of copper disks that could be separated thus altering its density between 3 and 9 g · cm−3 . The normal neutrino flux from π and K decays was reduced by about 3 orders of magnitude. Since charm is produced in pairs in proton interactions, the beam contains an approximately equal amount of neutrinos and antineutrinos, the only difference being due to π and K mesons decays occurring before these mesons are absorbed. Furthermore, because of the equal eνe and μνμ decay probabilties of charm an equal number of νe and νμ are present in the beam. In this context, it is interesting to note [18] that hadronic colliders, and in particular LHC, produce a large quantity of charm and beauty particles in the forward directions, resulting in two well collimated neutrino beams emerging from each interaction region.

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8.3 Detection Techniques Because of the small interaction cross section of neutrinos, neutrino detectors must be massive. The exception is detectors addressing coherent neutrino interactions for which the cross section is orders of magnitude larger than for other neutrino interactions and which will be addressed in Sect. 8.3.1. The nature of these massive detectors depends on the physics being addressed. It usually involves observing the resulting hadronic part of an interaction and, if a charged current interaction, the observation of a charged lepton. If the physics merely requires the measurement of the total neutrino energy, a calorimetric detector suffices. If individual particles must be measured, then a more sophisticated tracking device is needed. The measurement of a final state muon is usually accomplished in a fairly straightforward way with magnetized iron because of the muon penetrating nature. A final state electron is more difficult to measure, especially its charge, because of bremsstrahlung and showering as it propagates through material. Neutrino detectors must then necessarily be of several types. Techniques must be suitable to detect neutrinos of energies ranging from a few MeV to about a PeV. They must be fine-grained enough to measure electrons, identify individual particles and observe secondary vertices of τ ’s or charmed particles or heavy enough to produce large number of interactions using calorimetric techniques. It is evident that neutrino detectors use most of the detecting techniques used in particle physics. They will be outlined in the following sections.

8.3.1 Totally Active Scintillator Detectors Scintillator detectors can either use liquid or solid scintillator. If liquid is used the detector consists of either a single large tank or of tubes filled with liquid. Solid scintillator detectors usually consist of strips. The first neutrino detector [2] used by Reines and Cowan was intended to observe the interaction of reactor antineutrinos of a few MeV. The observation was made, as in subsequent reactor experiments, using the IBD reaction ν¯ e + p → e+ + n and using a detector consisting of liquid scintillator viewed by photomultipliers. In addition to observing light emitted from the positron annihilation, the neutron can be detected by also observing photons emitted by the neutron capture in the hydrogen of the scintillator. In order to enhance the neutron capture cross section, they added cadmium to the scintillator. They observed an excess of events when the reactor was in operation leading to the first detection of (anti)neutrino interactions and a subsequent Nobel prize. This technique is still being applied to this day [10], albeit with some refinements. Several recent experiments which will be described below, used it to search for ν¯ e oscillations to another flavour in the domain of the atmospheric m2 , 2.5 × 10−3 eV2 . Because of the low energy of reactor ν¯ e ’s, ν¯ μ ’s or ν¯ τ ’s that they potentially oscillate to cannot be observed through their charged current

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interactions since it is energetically impossible to produce μ’s or τ ’s. Oscillations can then only be observed through the disappearance technique resulting in a reduction and distortion of the expected ν¯e spectrum. Given the energy of reactor ν¯ e ’s (a few MeV) and the value of the atmospheric m2 , CHOOZ [19] was located 1000 m from a reactor complex in order to be near oscillation maximum. It used a single large tank of liquid scintillator and was subjected to a cosmic muon rate of 0.4 m−2 s−1 . One of the major backgrounds in this type of experiment is the background generated by cosmic ray muons. The first line of defense is to place the detector underground, at a depth of 300 m water equivalent (m.w.e) in the case of CHOOZ. A muon traversing the detector does not, in itself, simulate a signal event because the large amount of energy deposited can be well identified. However neutrons produced by muons traversing dead areas of the detector or the surrounding rock can elastically scatter on a proton, causing the proton and the subsequent neutron capture to simulate the signature of a reactor event. This background can be eliminated by vetoing on the passage of a nearby muon. In addition cosmic muons can produce long lived isotopes such as 6 He and 9 Li which subsequently can beta decay producing an electron and a neutron, thus simulating an antineutrino event. This background cannot be eliminated by vetoing on the passage of a muon because of the long lifetime of these decays (178 ms in the case of 9 Li) which would introduce an inordinate dead time. It must be estimated and subtracted. Palo Verde [20] was located at a shallower depth of 32 m.w.e. and chose to use acrylic cells filled with liquid scintillator. This extra segmentation was needed to reduce the larger muon induced neutron background caused by the larger cosmic muon flux of 22 m−2 s−1 at this depth. Instead of cadmium, these experiments have been using a 0.1% admixture of gadolinium with a large neutron absorption cross section leading to an 84% capture fraction. Absorption in gadolinium leads to the emission of gamma rays with a total energy of 8 MeV, within ∼30 μs and ∼6 cm of the positron annihilation, thus providing a well recognizable delayed coincidence. The CHOOZ target scintillator consisted of 50% by volume Norpar-15 [21] and IPB + hexanol (also 50% by volume). The wave-length shifters were p-PTP and bis-MSB (1 g/l). The gadolinium was introduced as a solution of Gd(NO3 )3 in hexanol. Because of oxygenation of the nitrate the 4 m light attenuation length in the scintillator decreased with time at a rate of (4.2 ± 0.4) · 10−3 per day. This required a careful monitoring of the scintillator transparency using calibration sources. The light yield was 5300 photons/MeV. The observation of a modification of the expected ν¯e spectrum necessitates a very precise knowledge of the antineutrino flux emitted by the reactor as well as of the antineutrino interaction cross section. They failed to observe a disappearance of antineutrinos and the limit set on this oscillation was governed by these sources of systematics uncertainty. In order to overcome these limitations more recent reactor oscillation experiments use a second identical detector located close to the reactor in order to measure the expected interaction rate before the neutrinos have a chance to oscillate. The detector used by one such experiment, Double Chooz [22], located at the same location as CHOOZ but using, in addition, a near detector placed at 410 m from the reactors, will be described as an example. The scintillator, amounting to

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Fig. 8.4 The Double Chooz detector

10 tons, is housed in a central tank, Fig. 8.4, consisting of a clear acrylic. It is surrounded by a gamma catcher consisting of undoped liquid scintillator housed in a second acrylic shell that provides additional gamma detection probability for interactions occurring near the boundary of the central tank. A third envelope consisting of stainless steel holds the photomultipliers that view the two scintillator volumes through the acrylic shells. A buffer consisting of mineral oil fills the space between the stainless steel shell and the acrylic shell of the gamma catcher. It serves the purpose of absorbing any radioactivity emitted by the photomultipliers. In order to veto on cosmic muons the photomultiplier shell is itself surrounded by yet another scintillator volume housed in a final outer shell that also holds a second set of photomultipliers viewing this inner veto layer. Lastly, planes of scintillator counters cover the ceiling of the detector cavern, to identify more muons that traverse the surrounding material and are a potential source of neutrons. The scintillator chosen for the target is a 20/80 mixture of phenyl-xylelethane (PXE)/dodecane with 0.1% gadolinium doping introduced as a dipivaloymethane molecule, Gd(dpm)3. This has demonstrated long term stability. With PPO and Bis-MSB as fluors the mixture has an attenuation length greater than 5 m at 450 nm and a light yield of 7000 photons/MeV resulting in 200 detected photoelectrons/MeV. The positron detection

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threshold is less than 700 keV, well below the threshold of 1.022 MeV of the inverse beta decay reaction. Calibration of the detector is required to determine, in both detectors, the efficiency for observing the inverse beta decay reaction, the energy scales for positrons, neutrons and gamma, the timing of the photomultipliers and the light transport properties. To this end gamma sources, neutron sources and laser light flashers are used and deployed throughout the detector volumes in order to map out the relevant parameters. In the target this is done with an articulated arm at the end of which is mounted the calibrating device, the position of which is determined by the length and azimuthal position of the arm. In the gamma catcher a guide tube into which a source can be inserted at the end of a wire is used. The geometry of the tube and the wire length determine the position of the source. Whereas Double Chooz was the first to report a hint for a non-zero θ13 , two experiments have since produced the best measurements of this angle. They use the same concept as Double Chooz but have used either a larger neutrino flux (RENO [23]) or more detectors and more flux (Daya Bay [24]). RENO, in South Korea, is exposed to the flux of 6 reactors in a row totalling 16.4 GWt h . Its far detector is 168 m underground and 1380 m from the central reactor whereas its near detector is 46 m underground and 290 m from the reactor line. Its inner target weighs 15.4 tons and is viewed by 340 photomultipliers. Daya Bay, in China, uses eight identical detectors and is located near three reactor complexes Daya Bay, Ling Ao I and II, a total of 17.4 GWt h . Its far detector hall is 324 m underground, 1540 m from Ling Ao and 1910 m from Daya Bay and houses four detectors. One near detector hall 363 m from the Daya Bay complex and another one about 500 m from the Ling Ao complex each house 2 detectors. Each detector includes a 20 ton neutrino target viewed by 192 8 photomultipliers. Their inner vetos are tanks of water in which Cerenkov light√ is viewed by photomultipliers. The Daya Bay energy resolution is σE /E = 7.5%/ E. Using a variety of radioactive sources they are able to determine the absolute neutrino energy scale to 1% and the relative energy scale between detectors to = kB T 2 C .

(19.5)

It is independent of the absorbed energy E, the thermal conductivity g of the heat link and the time constant τ . The above equation can intuitively be understood when assuming that the effective number of phonon modes in the detector is N = C/kB , the typical mean energy of one phonon is kB T and the rms fluctuation of one phonon is one. Then the mean square energy fluctuation is N(kB T )2 = kB T 2 C.

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For a practical cryogenic calorimeter the energy resolution is therefore given to first order by 1 EF W H M = 2.35ξ kB T 2 C ,

(19.6)

where ξ is a parameter which depends on the sensitivity and noise characteristics of the thermometer and can have values between 1.2 and 2.0. The best resolution obtained so far with cryogenic calorimeters is ∼2 eV at 6 keV. The use of superconductors as cryogenic particle detectors was motivated by the small binding energy 2 (order of meV) of the Cooper pairs. The breaking of a Cooper pair results in the creation of two excited electronic states the so-called quasi-particles. A particle traversing a superconductor produces quasi-particles and phonons. As long as the energy of the quasi-particles and the phonons is higher than 2, they break up more Cooper pairs and continue to produce quasi-particles until their energy falls below the threshold of 2. Particles which lose the energy E in an absorber produce ideally N = E/ quasi-particles. Thus the intrinsic energy resolution of a superconducting cryogenic calorimeter with  ≈ 1 meV is given by √ EF W H M = 2.35 F E∼2.6 eV

(19.7)

for a 6 keV X-ray assuming a Fano factor F = 0.2, which is representative for most superconductors and which takes the deviation from Poisson statistics in the generation of quasi-particles into account. For comparison, the energy resolution of a semiconductor device is typically: √ EF W H M = 2.35 wF E∼110 eV

(19.8)

for a 6 keV X-ray, where w is the average energy necessary to produce an electron hole pair. It has a typical value of w ≈ 3 eV. The Fano factor is F = 0.12 for Silicon. Because of the larger number of free charges a super-conducting device has a much better energy resolution. In Fig. 19.2 X-ray spectra obtained with a state of the art Si(Li) solid-state device (dashed line) and a cryogenic micro-calorimeter (solid line) using a Bi absorber and an Al-Ag bilayer superconducting transition edge thermometer are compared. The micro-calorimeter has been developed at the National Institute of Standards and Technology (NIST) in Boulder (USA) [42].

19.3 Phonon Sensors Phonons produced by a particle interaction in an absorber are far from thermal equilibrium. They must decay to lower energy phonons and become thermalized before the temperature rise T can be measured. The time required to thermalize

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877

TiN μcal EDS

μcal EDS Counts (0.16 eV bins)

N Kα

Si(Li) EDS

400

200

(Ti L Ti Lη)

Ti Lα1.2

Ti Lß3.4 Ti Lß1

0 360

400

440

480

520

Energy [eV] Fig. 19.2 TiN X-ray spectra obtained with a cryogenic micro-calorimeter (solid line) from the NIST group (see text) and with a state of the art Si(Li) solid-state device (dashed line) are compared. EDS stands for energy dispersive spectrometer. TiN is an interconnect and diffusion barrier material used in semiconductor industry

and the long pulse recovery time (τ = C/g) limits the counting rate of thermal calorimeters to a few Hz. The most commonly used phonon sensors are resistive thermometers, like semiconducting thermistors and superconducting transition edge sensors (TES), where the resistance changes as a function of temperature. These thermometers have Johnson noise and they are dissipative, since the resistance requires power to be read out, which in turn heats the calorimeter (Joule heating). However, the very high sensitivities of these calorimeters can outweigh to a large extent these disadvantages. There are also magnetic thermometers under development, which do not have readout power dissipation.

19.3.1 Semiconducting Thermistors A thermistor is a heavily doped semiconductor slightly below the metal insulator transition. Its conductivity at low temperatures can be described by a phonon assisted electron hopping mechanism between impurity sites. This process is also called “variable range hopping” (VRH) [44]. For temperatures between /10 mK . T0 12 and 4 K the resistance is expected to follow R(T ) = R0 exp ( T ) . This behavior is observed in doped Si and Ge thermistors. However, depending on the doping concentrations of the thermistor and the temperature range of its use, deviations from this behaviour have also been discovered. An important requirement for the fabrication of thermistors is to achieve a good doping homogeneity and reproducibility. Good uniformity of doping concentrations has been achieved either

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with ion implantation or with neutron transmutation doping (NTD). In the latter case, thermal neutrons from a reactor are captured by nuclei which transform into isotopes. These can then be the donors or acceptors for the semiconductor. NTD Ge thermistors are frequently used because of their reproducibility and their uniformity in doping density. Furthermore they are easy to handle and commercially available. It is convenient to define a dimensionless sensitivity of the thermometer: α≡

T dR d log R = . d log T R dT

(19.9)

The energy resolution of these devices is primarily driven by the heat capacity C of the absorber, the sensitivity of the thermometer α, the Joule heating, the Johnson noise of the load resistor and the amplifier noise. The bias current through the resistor can be optimized in such a way that it is kept high enough to provide a suitable voltage signal and low enough to minimize the Joule heating. If also the Johnson noise and the amplifier noise can be kept sufficiently low, the energy resolution of an ideal calorimeter can be described to first order by Eq. (19.6), where ξ is approximately 5(1/α)1/2 [39]. For large values of α the energy resolution can be even much better than the magnitude of the thermodynamic fluctuations provided no power is dissipated by the temperature measurement of the sensor. Semiconducting thermistors have typically α values between 6 and 10, while superconducting transition edge sensors (TES) have values which are two orders of magnitude higher. A detailed description of the noise behavior and the energy resolution of cryogenic detectors can be found in [39–41].

19.3.2 Superconducting Transition Edge Sensors (TES) A frequently used phonon sensor is the so-called transition edge sensor (TES). It consists of a very thin superconducting film or strip which is operated at a temperature in the narrow transition region between the superconducting and the normal phase, where its resistance changes between zero and its normal value RN , as shown in Fig. 19.3a. TES sensors are usually attached to an absorber, but they can also be used as absorber and sensor at the same time. The very strong dependence of the resistance change on temperature, which can be expressed in the dimensionless parameter α of Eq. (19.9), makes the TES calorimeter sensitive to very small input energies. Superconducting strips with low Tc can have α values as high as 1000. This requires very high temperature stability. The Munich group has developed one of the first TES sensors, which was made from tungsten with a transition temperature of 15 mK [45]. The TES sensor can be operated in two different modes: the current and the voltage biased mode. In a current biased mode of operation a constant current is fed through the readout circuit as shown in Fig. 19.3b. A particle interaction in the absorber causes a temperature rise and a corresponding increase of the

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Resistance

TC SQUID

RN

ΔR

L

R(T)

Top

RL

a

Temperature ΔT

b

Iext

Fig. 19.3 (a) The temperature versus resistance diagram of a superconducting strip close to the transition temperature Tc is shown. (b) The dc-SQUID readout of a transition edge sensor is shown

resistance R(T ) of the attached TES sensor. The change of the resistance forces more current through the parallel branch of the circuit, inducing a magnetic flux change in L which is measured with high sensitivity by a superconducting quantum interference device (SQUID). However, in this mode Joule heating by the current through the sensor and small fluctuations in the bath temperature can prevent to achieve good detector performances. To solve this problem K.D. Irwin [46] has developed a so-called auto-biasing electro-thermal feedback system (ETF), which works like a thermal equivalent to an operation amplifier and keeps the temperature of the superconducting strip at a constant value within its transition region. When operating the transition edge sensor in a voltage biased mode (VB ), a temperature rise in the sensor causes an increase in its resistance and a corresponding decrease in the current through the sensor, which results in a decrease of the Joule heating (VB · I ). The feedback uses the decrease of the Joule heating to bring the temperature of the strip back to the constant operating value. Thus the 0device is self-calibrating. The deposited energy in the absorber is given by E = VB I (t)dt. It can directly be determined from the bias voltage and the integral of the current change. The use of SQUID current amplifiers allows for an easy impedance matching to the low resistance sensors and opens the possibility to multiplex the read out of large arrays of TES detectors. Another advantage of ETF is that in large pixel arrays the individual channels are self-calibrating and temperature regulated. Most important, ETF shortens the pulse duration time of TES by two orders of magnitude compared to thermistor devices allowing for higher count rates of the order of 500 Hz. The intrinsic energy resolution for an ideal TES calorimeter is given by Eq. (19.6) with ξ = 2(1/α)1/2 (n/2)1/4, where n is a parameter which depends on the thermal impedance between the absorber (phonons) and the electrons in the superconducting film [46, 47]. For thin films and at low temperatures the electron-phonon decoupling dominates in the film and n is equal to 5. The best reported energy resolutions with TES devices so far are a little below 2 eV at 6 keV.

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The observed transition width of TES T in the presence of a typical bias current is of the order of a few mK. Large bias currents usually lead to transition broadenings due to Joule heating and self-induced magnetic fields. In order to achieve best performance of TES in terms of energy resolution or response time for certain applications, specific superconducting materials have to be selected. Both superconductors of type I and type II qualify in principle. However, the physics of the phase transition influences the noise behavior, the bias current capability and the sensitivity to magnetic field of the TES. Sensors made from high temperature superconductors have a much lower sensitivity than low temperature superconductors due to the larger gap energies  and heat capacities C. Thermal sensors made from strips of Al (with Tc = 1.140 K), Ti (0.39 K), Mo (0.92 K), W (0.012 K) and Ir (0.140 K) have been used. Ti and W sensors have been developed in early dark matter detectors [48–51]. But also other transition edge sensors, made from proximity bi-layers such as Al/Ag, Al/Cu, Ir/Au, Mo/Au, Mo/Cu, Ti/Au, or multi-layers such as Al/Ti/Au [56], have been developed to cover transition temperatures in the range between 15 and 150 mK. Although not all of these combinations are chemically stable, good detector performances have been obtained with Ir/Au bi-layers [52, 53] at transition temperatures near 30 mK. Methods to calculate bi-layer Tc can be found in [54, 55]. Another method to suppress the Tc of a superconducting film is to dope it with magnetic ions, like for example Fe ( 2Δ1

S

Ef

S´ S1 eV

S´

S S2

Fig. 19.4 The processes in a superconducting tunnel junction (ST) are illustrated

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or X-ray interaction in film S1 the quasi-particle density is increased. This will lead to an increase of a net quasi-particle transfer from S1 to S2 and consequently to an increase of the tunneling current. However, not all quasi-particles will reach and pass through the junction barrier. Depending on the geometry and structure of the junction there will be losses. Quasi-particles can recombine to Cooper pairs radiating phonons as consequence of the relaxation process. If the phonon energy is high enough 0 > 21 to break new Cooper pairs, this process can lead to quasi-particle multiplication enhancing the signal output of the STJ. If, however, the phonon energy is below the energy threshold for breaking a Cooper pair 0 < 21 the quasi-particle will be lost and will not contribute to the signal. Quasi-particles will also be lost when they diffuse out of the overlap region of the junction films into the current leads instead of crossing the junction barrier. N. Booth [68] proposed a scheme which allows to recover some of these losses by quasi-particle trapping and in some cases quasi-particle multiplication. Quasi-particle trapping can be achieved by introducing bi-layers of superconducting materials (S(1 ) and S’(2 )) with different gap energies 2 < 1 [68]. For example, an X-ray absorbed in the superconductor S produces phonons of energy 0 > 21 breaking a number of Cooper pairs. Some of the produced quasi-particles diffuse to the superconducting film S’ of the STJ with a smaller gap energy 2 . By falling in that trap they relax to lower energies by emitting phonons, which could generate additional quasi-particles in the film S’ (quasi-particle multiplication) if their energy is larger than 22 . However, the relaxed quasi-particles cannot diffuse back into the superconductor S because of their lower energy. They are trapped in S’ and will eventually tunnel through the STJ, contributing to the signal with the tunneling current ia . In order for quasi-particle trapping to be effective superconducting absorber materials with long quasi-particle lifetimes have to be selected (for example Al). Back tunneling, the so-called Gray effect, is also enhancing the signal [69]. In this Cooper pair mediated process a quasi-particle in film S2 recombines to form a Cooper pair at the expense of a Cooper pair in film S1. In this case the quasi-particle current ib is also running in the direction of decreasing potential. Thus both excess quasiparticle currents ia and ib have the same sign. This feature allows to record signals from X-rays absorbed in either superconducting films S1 or S2 with the same sign. However, their signal shapes may not necessarily be the same due to different quasiparticle and tunneling losses in the two films. There are two other ways of electrical transport through the tunnel barrier which need to be suppressed when the STJ is used as a particle detector. One is the so-called dc Josephson current of Cooper pairs through the tunnel barrier. This current can be suppressed by applying a magnetic field of the order of a few Gauss parallel to the insulating barrier. The second is the tunnel current generated by thermally excited excess quasi-particles. The number density of these quasi-particles is decreasing with decreasing temperature according to Nt h ∼ T 1/2 exp(−/kB T ). In order to obtain a significant signal to background ratio the operating temperature of a STJ detector should be typically lower than 0.1 Tc . The intrinsic energy resolution of the excess quasi-particles in a STJ device is given by Eq. (19.7), where  has to be replaced by , the effective energy needed to create one excited state. It turns out that  ∼ 1.7  for Sn and Nb superconductors,

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reflecting the fact that only a fraction of the absorbed energy is transfered into quasiparticles [70]. The number of quasi-particles generated in the STJ by an energy loss E of a particle in the superconductor is thus N = E/. For a Nb superconductor with a Fano factor F = 0.2 and  = 2.5 meV one would expect from Eq. (19.7) an energy resolution of 4 eV at 6 keV. However, the best resolution observed so far is 12 eV at 6 keV. In order to estimate a more realistic energy resolution, quasi-particle loss and gain processes have to be taken into account. The two most important parameters driving the energy resolution of the STJ are the tunneling rate t ≡ τt−1 and the thermal recombination rate r ≡ τr−1 . The temperature dependence of the thermal recombination rate is given by τr−1 (T )

=

√ τ0−1 π



2 k B Tc

5/2

' (  T exp (− ) Tc kB T

(19.11)

where τ0 is the characteristic time of a superconductor. It has the values τ0 = 2.3 ns for Sn, τ0 = 438 ns for Al and τ0 = 0.15 ns for Nb [71]. The recombination rate r ≡ τr−1 can be minimized when operating the detector at sufficiently low temperatures, typically at 0.1 Tc , where the number of thermally excited quasi-particles is very small. The tunneling rate of a symmetric STJ is given by de Korte et al. [72]  + eVb τt−1 = (4 Rnorm · e2 · N0 · A · d)−1 1 ( + eVb )2 − 2

(19.12)

where Rnorm is the normal-conducting resistance of the junction, N0 is the density of states of one spin at the Fermi energy, A is the junction overlap area, d the thickness of the corresponding film and Vb the bias voltage of the STJ. In practice, the tunneling time has to be shorter than the quasi-particle lifetime. For a 100×100 μm2 Nb-Al tunnel junctions with Rnorm = 15 m a tunneling time of τt = 220 ns and a recombination time of τr = 4.2 μs has been measured [66]. The recombination time was determined from the decay time of the current pulse. Thus quasi-particles tunneled on average 19 times. In order to achieve even shorter tunneling times, one would have to try to further reduce Rnorm . However, there is a fabricational limit avoiding micro-shorts in the insulator between the superconducting films. The STJ counting rate capability is determined by the pulse recovery time, which depends on the quasi-particle recombination time and can have values between several μs and up to ∼50 μs. Typical count rates of STJs are 104 Hz. An order of magnitude higher count rates can still be achieved, but not without losses in energy resolution. The total quasi-particle charge collected in the STJ is to first order given by Q = Q0

t d

(19.13)

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with Q0 = Ne and d = 2r + loss the total quasi-particle loss rate. The factor 2 in the recombination rate takes into account the loss of two excited electronic states and loss stands for all the other quasi-particle losses, mainly due to diffusion. These effects can be parametrized into an effective Fano factor which is added into the equation for the energy resolution 1 EF W H M = 2.35 (F + G)E .

(19.14)

For a symmetric tunnel junction with equal tunneling probabilities on both sides the G factor is given by G = (1 + 1/n) ¯ with n¯ = Q/Q0 = t / d [66, 67]. It emphasizes the importance of a large tunneling rate. Still the energy resolution in Eq. (19.14) is only approximative since it neglects gain factors like quasiparticle multiplication due to relaxation phonons and loss factors due to cancellation currents, which becomes important at low bias voltage. From Eq. (19.12) it is clear that in order to achieve a high tunnel rate the STJ detector has to be made from very thin films with a small area A. These dimensions also determine the capacitance which should be kept as small as possible in order not to degrade the detector rise time and the signal to noise ratio. For the very thin films the quantum efficiencies at X-ray energies are very low. This can be changed by separating the absorber and detector functions in fabricating devices with a larger size superconducting absorber as substrate to a STJ. Quasi-particle trapping will be achieved when choosing substrate materials with a higher energy gap with respect to the junction. Quasi-particle trapping was first demonstrated using a Sn absorber (975 × 150 × 0.25 μm3 ) with an energy gap of Sn = 0.58 meV and with an Al-Al2 O3 -Al STJ at each end of the absorber [65]. Quasi-particles generated by an event in the Sn absorber diffuse into the aluminum junctions, where they stay trapped because of the smaller energy gap Al = 0.18 meV of Al with respect to Sn. The excess quasiparticle tunnel current was then measured with the two Al STJs. In order to prevent diffusion losses out of the Sn absorber the common contact leads to the STJs and to the Sn absorber where made from Pb, which has an even higher energy gap of Pb = 1.34 meV. It turned out that more than 99.6% of the quasi-particles which tried to diffuse out of the absorber where rejected at the Pb barrier and hence confined to the absorber. Currently the best energy resolution of 12 eV at 6 keV has been achieved with a single Al-Al2 O3 -Al STJ using a superconducting Pb absorber (90 × 90 × 1.3 μm3 ) with an absorption efficiency of ≈50% [73]. It turns out that Al is an ideal material for STJ because it allows to fabricate a very uniform layer of the tunnel barrier and has a very long quasi-particle lifetime. These are features which are essential for a high performance STJ. A very good description of the physics and applications of STJ detectors can be found in [67]. Arrays of STJs have been developed for astronomical observations and other practical applications as discussed in the chapters below. The Naples collaboration [76] produced an array of circular shaped STJs [76, 77]. This device allows the operation of STJs without external magnetic field. Position sensitive devices have been developed for reading out large pixel devices [65, 74, 78].

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19.4.2 Microwave Kinetic Inductance Detector A new detector concept, called microwave kinetic inductance detector (MKID) has been introduced with the aim to develop multi-pixel array cameras for X-ray and single photon detection [79, 80]. MKID is, like STJ and SSG, a non-equilibrium detector which is based on Cooper pair breaking and the production of quasiparticles. The basic element of the device consists of a thin superconducting film, which is part of a transmission line resonator. The principle of detection is shown in Fig. 19.5, taken from [79]: A photon absorbed in a superconducting film will break up Cooper pairs and produce quasi-particles (a). The increase of quasiparticle density will affect the electrical conductivity and thus change the inductive surface impedance of the superconducting film, which is used as part of a transmission line resonator (b). At resonance, this will change the amplitude (c) and the transmission phase of the resonator (d).

Fig. 19.5 The basic operation of a MKID (Microwave Kinetic Inductance Detector) is shown

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The change in the transmission phase is proportional to the produced number of quasi-particles and thus to the photon energy. First measurements with an X-ray source yielded an energy resolution of 11 eV at 6 keV. MKID detectors find many applications where a large number of pixels are demanded. As compared to other devices multiplexing can be realized rather easy by coupling an array of many resonators with slightly different resonance frequencies to a common transmission line. A single amplifier is needed to amplify the signals from a large number of detectors. Due to its interesting features MKID is under development for many applications in ultraviolet, optical and infrared imaging [80].

19.4.3 Superheated Superconducting Granules (SSG) Superheated superconducting granules (SSG) have been developed for X-ray imaging, transition radiation, dark matter as well as solar and reactor neutrino detection [10, 94]. A SSG detector consists of billions of small grains (typically 30 μm in diameter), diluted in a dielectric material (e.g. Teflon) with a volume filling factor of typically 10%. The detector is operated in an external magnetic field. Metastable type-1 superconductors (e.g. Sn, Zn, Al, Ta) are used, since their phase transitions from the metastable superconducting state to the normal-conducting state are sudden (in the order of 100 ns) allowing for a fast time correlation between SSG signals and those of other detectors. Its energy threshold is adjustable by setting the external magnetic field at a certain value H just below the phase transition border. The phase diagram of a type-1 superconductor is schematically shown in Fig. 19.6, where Hsh is the superheating field, Hsc is the supercooling field and Hc is the critical thermodynamic field which is approximately given by Hc (T ) = Hc (0)(1 − ( TTc )2 ). The region below Hsc is the superconducting and above Hsh the normal-conducting phase, while the region between the two is the so-called meta-stable phase, which is characteristic for superconductors of type-1. In order to keep the heat capacity as low as possible the SSG detector is operated at a temperature much below the critical temperature Tc at typically T0 ≈ 100 mK. Particles interacting in a granule produce quasi-particles. While spreading over the volume of the granule the quasi-particles are losing energy via electron-phonon interactions, thereby globally heating the granule up to a point where it may undergo a sudden phase transition (granule flip). The temperature change experienced by the 3E granule is T = 4πcr 3 , with E the energy loss of the particle in the grain, c the specific heat and r the radius of the grain. The phase transition of a single grain can be detected by a pickup coil which measures the magnetic flux change  due to the disappearance of the Ochsenfeld-Meissner effect. In case of a single grain located in the center of the pickup coil the flux change is given by  = 2πBn √

r3 4R 2 + l 2

(19.15)

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H

ΔH

ΔT

Hc (0)

Hsh

Hc

Hsc

T0

T1

Tc

T

Fig. 19.6 The phase-diagram of a superconductor type1 is shown. Hsh is the superheating field, Hsc is the supercooling field and Hc is the critical thermodynamic field

with B the applied magnetic field, n the number of windings, R the radius and l the length of the pickup coil. It should be noted that one coil may contain a very large number of grains. If the flipping √time τ is small compared to the characteristic time of the readout circuit (τ  2π LC) the flux change induces a voltage pulse in the pick-up coil V (t) =

 −t /2RC 1 1 e − sin(ωt) , ω2 = ωLC LC (2RC)2

(19.16)

with ω, L, R and C being parameters of the pick-up circuit. A detailed description of a readout concept using conventional pick-up coils and electronics including noise estimation is given in [81, 82]. Besides conventional readout coils more sensitive Superconducting Quantum Interference Devices (SQUID) were introduced [83–85]. The SQUID readout allows the detection of single flip signals from smaller size granules and/or the usage of larger size pickup coils. Granules of 20 μm diameter were measured in a large size prototype [85]. Small spherical grains can be produced at low cost by industry using a fine powder gas atomization technique. Since after fabrication the grains are not of a uniform diameter, they have to be sieved to select the desired size. A grain size selection within ±2 μm was achieved.

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The Bern Collaboration has built and operated a dark matter SSG detector, named ORPHEUS, which consisted of 0.45 kg of spherical Sn granules with a diameter of ≈30 μm [81]. The detector was read out by 56 conventional pick-up coils, each 6.8 cm long and 1.8 cm in diameter. Each pick-up coil contained ≈80 million granules. The phase transition of each individual grain could be detected with a typical signal to noise ratio of better than 10. The principle to detect small nuclear recoil energies with SSG was successfully tested prior to the construction of the ORPHEUS detector in a neutron beam of the Paul Scherrer Institute (Villigen, Switzerland) [86]. The special cryogenics required for the ORPHEUS detector is described in [87]. The detector is located in the underground facility of the University Bern with an overburden of 70 meter water equivalent (m.w.e.). In its first phase the ORPHEUS dark matter experiment did not reach the sensitivity of other experiments employing cryogenic detectors, as described below. Further improvements on the superconducting behavior of the granules and on the local shielding are necessary. SSG is a threshold detector. Its resolution depends on the sharpness δH /H , respectively δT /T , of the phase transition. It was found that the phase transition smearing depends on the production process of the grains. Industrially produced grains using the atomization technique exhibited a smearing of δH /H ∼ 20%. By using planar arrays of regularly spaced superheated superconducting microstructures which were produced by various sputtering and evaporation techniques the transition smearing could be reduced to about 2% [88–92]. The improvement of the phase transition smearing is one of the most important developments for future applications of SSG detectors. It looks promising that large quantities of planar arrays can be produced industrially [92]. There is a mechanism by which the energy transfered to a grain can be measured directly. If the grain is held in a temperature bath just below the Hsc boundary and the energy (heat) transfer to the grain is large enough to cross the meta-stable region to become normal-conducting, it will after some time cool down again to the bath temperature and become superconducting again. During this process the granule will provide a “flip” signal when crossing the Hsh border, and an opposite polarity “flop” signal when crossing the Hsc border. The elapsed time between the flip and the flop signal is a measure of the deposited energy in the grain. This effect has been demonstrated with a 11 μm Sn grain bombarded with α particles [93]. It offers the possibility to build an energy resolving and self-recovering SSG detector. The practical realization of a large SSG detector is still very challenging. Nevertheless, the detector principle offers several unique features: (a) The large list of suitable type-1 superconductor materials allows to optimize SSG for specific applications. (b) Very low energy thresholds (eV) can be achieved. (c) The inductive readout does not dissipate any power into the grains. Therefore the sensitivity of SSG is essentially determined by the grain size and the specific heat of the grain material.

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(d) The sudden phase transitions are beneficial for coincident timing with other signals. Generally speaking, SSG detectors are among the most sensitive devices to detect very low energy transfers, i.e. nuclear recoils. A detailed description of SSG can be found in [10, 94, 95].

19.5 Physics with Cryogenic Detectors 19.5.1 Direct Dark Matter Detection Among the most challenging puzzles in physics and cosmology is the existence of dark matter and dark energy. Dark matter, which was first inferred by Fritz Zwicky in 1933 [96], shows its presence by gravitational interaction with ordinary matter. It holds numerous galaxies together in large clusters and it keeps stars rotating with practically constant velocities around the centers of spiral galaxies. Dark energy, which was discovered by the Supernovae type 1a surveys in 1998 [97, 98], is driven by a repulsive force quite in contrast to the attractive gravitational force and causes the universe to expand with acceleration. The most recent information about the matter/energy content of the universe was gained from the Cosmic Microwave Background radiation (CMB) measurements by the Planck satellite [99]. According to these observations the universe contains 69.4% dark energy, 30.6% matter (including baryonic and dark matter) and 4.8% baryonic matter in form of atoms. The true nature of the dark energy and the dark matter, which fills about 95% of the universe, is still unknown. The direct detection of the dark energy, which is related to Einstein’s cosmological constant, seems not to be in reach with present technologies. However, the direct detection of dark matter, if it exists in form of particles, is encouraged by the large expected particle flux which can be deduced under the following assumptions. In an isothermal dark matter halo model the velocity of particles in our galaxy is given by a Maxwell Boltzmann distribution with an average value of < v >= 230 km s−1 and an upper cutoff value of 575 km s−1 corresponding to the escape velocity. The dark matter halo density in our solar neighborhood is estimated to be ρ = 0.3 GeV c−2 cm−3 . From that one expects a flux of  = ρ < v > /mχ ∼ 7 · 106/mχ cm−2 s−1 with mχ the mass of the dark matter particle in GeV c−2 . However, since neither the mass nor the interaction cross section of these particles are known one is forced to explore a very large parameter space, which requires very sensitive and efficient detection systems. The most prominent candidates for the dark matter are: massive neutrinos, WIMPs (weakly interacting massive particles) and axions. Neutrinos are among the most abundant particles in the universe, but their masses seem to be too small to contribute significantly to the missing mass. Neutrinos being relativistic at freeze out are free streaming particles, which cluster preferentially at very large scales. Therefore massive neutrinos would enhance large-scale and suppress small-scale structure formations. From hot dark matter and cold dark matter model calculations

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fitting the power spectrum obtained from Large Scale Structure (LSS) surveys one obtains a value for the ratio neutrino density to matter density ν / m . From this value and CMB an upper limit for the sum of the neutrino  m obtained from masses mν ≤ 0.234 eV c−2 can be derived. However, this and the results from direct neutrino mass experiments, as described below, indicate, that neutrinos have a mass to low to qualify for the dark matter. The introduction of axions was not motivated by cosmological considerations, but rather to solve the charge conjugation and Parity violation (CP) problem in Quantum Chromo-Dynamics (QCD) [100]. Nevertheless axions would be produced abundantly during the QCD phase transition in the early universe when hadrons were formed from quarks and gluons. A recent review of axion searches can be found in [101]. The most favored candidate for a WIMP is the neutralino, which is predicted by some Super Symmetric Theories (SUSY) to be the lightest stable SUSY particle. If the neutralino were to be discovered by the Large Hadron Collider (LHC) at CERN, it still would need to be confirmed as a dark matter candidate by direct detection experiments. However, up to now no sign of SUSY-particles has been observed at the LHC [102]. In the following the WIMP searches with some of the most advanced cryogenic detectors are described. The direct detection of WIMPs is based on the measurement of nuclear recoils in elastic WIMP scattering processes. In the case of neutralinos, spin-independent coherent scatterings as well as spin-dependent scatterings are possible. The expressions for the corresponding cross sections can be found in [103, 104]. In order to obtain good detection efficiencies, devices with high sensitivity to low nuclear recoil energies (eV) are needed. WIMP detectors can be categorized in conventional and cryogenic devices. Most of the conventional WIMP detectors use NaI, Ge crystals, liquid Xenon (LXe) or liquid Argon (LAr). These devices have the advantage that large detector masses (∼ton) can be employed, which makes them sensitive to annual modulations of the WIMP signal owing to the movement of the earth with respect to the dark halo rest frame. Annual modulation, if observed, would provide strong evidence for a WIMP signal, assuming it is not faked by spurious modulated background signals. However, due to quenching of the ionization signals, conventional detectors have lower nuclear recoil detection efficiencies than cryogenic devices. Cryogenic detectors are able to measure small recoil energies with high efficiency because they measure the total deposited energy in form of ionization and heat. They can be made of many different materials, like Ge, Si, TeO2 , sapphire (Al2 O3 ), LiF, CaWO4 and BGO, including superconductors like Sn, Zn, Al, etc. This turns out to be an advantage for the WIMP search, since for a given WIMP mass the resulting recoil spectra are characteristically different for detectors with different materials, a feature which helps to effectively discriminate a WIMP signal against background. If the atomic mass of the detector is matched to the WIMP mass better sensitivity can be obtained due to the larger recoil energies. In comparison to conventional detectors, however, cryogenic detectors are so far rather limited in target mass (∼kg).

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Dark matter detectors have to be operated in deep underground laboratories in order to be screened from cosmic-ray background. In addition they need to be shielded locally against radioactivity from surrounding rocks and materials. The shielding as well as the detector itself has to be fabricated from radio-poor materials, which turns out to be rather expensive and limited in its effectiveness. Nevertheless, cryogenic detectors are capable of active background recognition, which allows to discriminate between signals from background minimum ionizing particles, i.e. Compton electrons, and signals from genuine nuclear recoils by a simultaneous but separate measurement of phonons and ionization (or photons) in each event. For the same deposited energy the ionization (or photon) signal from nuclear recoils is highly quenched compared to signals from electrons. The dual phonon-ionization detection method, which was first sugested by Sadoulet [105] and further developed by the CDMS and EDELWEISS collaborations, increases the sensitivity for WIMP detection considerably. A similar idea using scintillating crystals as absorbers and simultaneous phonon-photon detection was introduced by Gonzales-Mestres and Perret-Galix [106] and further developed by the ROSEBUD [107] and CRESST II [108] collaborations. The principle of the method is demonstrated in the scatterplot of Fig. 19.7, taken from [108]. It shows the energy equivalent of the pulse heights measured in the light detector versus those measured in the phonon detector. The scintillating CaWO4 crystal absorber was irradiated with photons and electrons (using Cobalt and Strontium sources respectively) as well as with neutrons (using an Americium-Beryllium source). The photon lines visible in Fig. 19.7 were used

140

Pulse height in light detector [keV]

120 e

100 80 60

n

40 20 0 0

20

40

60

80

100

120

140

Pulse height in phonon detector [keV] Fig. 19.7 The energy equivalent of the pulse heights measured with the light detector versus those in the phonon detector under electron, photon (e) and neutron (n) irradiation are shown

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for the energy calibration in both the light and the phonon detector. The upper band in Fig. 19.7 shows electron recoils (e) and the lower band the nuclear recoils (n). Above an energy of 15 keV 99.7% of the electron recoils can be recognized and clearly distinguished from the nuclear recoils. Active background rejection was also practiced with the ORPHEUS SSG dark matter detector, since minimum ionizing particles cause many granules to flip, while WIMPs cause only one granule to flip (flip meaning a transition from superconducting to normal state) [81]. In the following some of the most sensitive cryogenic WIMP detectors in operation are described. The CDMS experiment [109] is located at the Soudan Underground Laboratory, USA, with an overburden of 2090 meter water equivalent (m.w.e.). In an early phase of the experiment the cryogenic detectors consisted of 4 towers of 250 g Ge absorbers which where read out by NTD germanium thermistors, so called Berkeley Large Ionization and Phonon (BLIP) detectors, and two towers of 100 g Si absorbers, which where read out by TES sensors, the so-called Z-sensitive Ionization and Phonon based (ZIP) detectors. The ZIP detectors utilize tungsten aluminum Quasi-particle trapping assisted Electrothermal feedback Transition edge sensors (QET). This type of sensor covers a large area of the Si absorber with aluminum phonon collector pads, where phonons are absorbed by breaking Cooper pairs and forming quasi-particles. The quasi-particles are trapped into a meander of tungsten strips which are used as transition edge sensors. The release of the quasi-particle energy in the tungsten strips increases their resistance, which will be observed as a current change in L detected with a SQUID as indicated in Fig. 19.3b. The transition edge device is voltage biased to take advantage of the electrothermal feedback (ETF). The signal pulses of the ZIP detector have rise times of a few μs and fall times of about 50 μs. They are much faster than the signals of the BLIP detector since the ZIP detectors are sensitive to the more energetic non thermal phonons. Their sensitivity to non thermal phonons and the pad structure of the sensors at the surface of the crystal allows for a localization of the event in the x-y plane. A separate circuit collects ionization charges, which are drifted by an electric field of 3 V/cm and collected on two concentric electrodes mounted on opposite sides of the absorber. The ratio of the ionization pulse height to the phonon pulse height versus the pulse height of the phonon detector allows to discriminate nuclear from electron recoils with a rejection factor better than 104 and with full nuclear recoil detection efficiency above 10 keV. In order to further improve the sensitivity of the experiment CDMS II is operating at present 19 Ge (250 g each) and 11 Si (100 g each) ZIP type detectors in the Soudan Underground Laboratory at a temperature of about 40 mK. Each detector is 7.62 cm in diameter and 1 cm thick. Limits on the direct detection of WIMPs obtained with the Ge and Si detectors are published in [110] and [111] respectively. The CDMSlite (Cryogenic Dark Matter Search low ionization threshold experiment) uses the Neganov-Luke effect, which leads to an amplification of the phonon signal and allows for lower energy thresholds (56 eV) to be reached [112–114]. In this mode a large detector bias voltage is applied to amplify the phonon signals produced by drifting quasi particle charges. This opens the possibility to extend the WIMP search to masses well below 10 GeV c−2 .

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Recent results on low mass WIMP searches for spin independent and spin dependent interactions are published in [115]. The EDELWEISS experiment [116], which is located in the Frejus tunnel (4800 m.w.e.), South of France, uses a technique similar to the CDMS BLIP detectors. It consists of 3 towers of 320 g Ge absorbers which are read out by NTD germanium thermistors. For the ionization measurement the detectors are equipped with Al electrodes which are directly sputtered on the Ge absorber crystal. For some data taking runs the EDELWEISS group used towers with amorphous Ge and Si films under the Al electrodes. More data were collected between 2005 and 2011 with the EDELWEISS II detector which contains an array of ten cryogenic Ge detectors with a mass of 400 g each [117]. As an upgrade of EDELWEISS II the collaboration developed EDELWEISS III with 36 FID (Fully Inter-Digital) detectors based on cylindrical Ge crystals with a mass of about 800 g each operating at 18 mK [118]. Early experience with the ionization measurements showed a severe limitation of the background separation capability due to insufficient charge collection of surface events. The effect can be attributed to a plasma screening of the external electric field of the electrodes. As a result surface interactions of electrons can fake nuclear recoil events. One way to solve the problem was developed by the Berkeley group by sputtering films of amorphous Si or Ge on the absorber surface before deposition of the Al electrodes [119]. Due to the modified energy band gap of the amorphous layer the charge collection efficiency was largely improved. Fast phonon detectors like ZIP allow to identify surface events by measuring the relative timing between the phonon and ionization signals as demonstrated by the CDMS experiment [120]. The surface event problem completely disappears when using dual phonon-photon detection. This method was chosen by the CRESST II collaboration. The CRESST experiment [121] is located in the Gran Sasso Underground Laboratory (3800 m.w.e.) north of Rome, Italy. It uses scintillating CaWO4 crystals as absorber material. The detector structure can hold 33 modules of absorber which can be individually mounted and dismounted. Each module weights about 300 g. The detector operates at about 10 mK. The phonon signal from the CaWO4 crystal is read by a superconducting tungsten TES thermometer and the photon signal by a separate but nearby cryogenic light detector, which consists of a silicon wafer with a tungsten TES thermometer. For an effective background discrimination the light detector has to be very efficient. This was achieved by applying an electric field to the silicon crystal leading to an amplification of the thermal signal due to the Neganov-Luke effect [122]. The time constant of the emission of scintillation photons from CaWO4 at mK temperatures is of the order of ms, which requires a long thermal relaxation time for the light detector. The characteristics of the background rejection power depends on the knowledge of the quenching factor, which is the reduction factor of the light output of the nuclear recoil event relative to an electron event. These quenching factors were measured by the CRESST collaboration for various recoiling nuclei in CaWO4 in a separate experiment [123]. The knowledge of these quenching factors would allow in principle to identify WIMP interactions with different nuclei in the CaWO4 crystal. This method seems very promising, not only for identifying the background but also the quantum

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numbers of the WIMP candidates. The three types of nuclei in CaWO4 together with a low nuclear recoil energy threshold of 300 eV allowed CRESST II to extend the dark matter search with high sensitivity into a mass region below 10 GeV c−2 [124]. Besides the dark matter search the CRESST collaboration is developing a cryogenic detector to measure coherent neutrino nucleus scattering [125]. This process is predicted by the Standard Model (SM), but has been unobserved so far. If successful, a possible application could be the real time monitoring of nearby nuclear power plants. With a small size prototype cryogenic Saphire detector with a weight of 0.5 g a recoil energy threshold of 20 eV was achieved [126]. Nevertheless, the first detection of coherent neutrino scattering was reported from the COHERENT collaboration only recently [127]. They measured neutrinoinduced recoils with conventional scintillating CsI(Na) crystals with a weight of 14.5 kg. Their experiment was located in a basement under the Oak Ridge National Laboratory Spallation Neutron Source. The experimental results are usually presented as exclusion plots, which show the WIMP-nucleon cross section versus the WIMP mass. They are derived from the expected nuclear recoil spectrum for a given set of parameters [104]: σ0 ρχ 2 dR = F (E) dE 2μ2 mχ

υmax

f (υ) dυ υ

(19.17)

υmin

with mχ the mass of the WIMP, μ the reduced mass of the WIMP-nucleus system, σ0 the total elastic cross section at zero momentum transfer, ρχ = 0.3 GeV cm−3 the dark matter halo density in the solar neighborhood, F (E) the nuclear form factor, f (υ) an assumed isothermal Maxwell-Boltzmann velocity distribution of the 1 WIMPs in the halo, υmin = EmN /2μ2 the minimum velocity which contributes to the recoil energy E, and υmax = 575 km s−1 the escape velocity from the halo. The recoil energy E is given by E = μ2 υ 2 (1 − cos θ )/mN , with mN the mass of the nucleus, υ the velocity of the WIMP, and θ the scattering angle in the centre of mass system. The expected nuclear recoil spectrum for interactions with WIMPs of a given mass will then be folded with the detector response, which was obtained experimentally from calibration measurements with neutron sources or in neutron beams. From a maximum likelihood analysis, an upper limit cross section value (90% C.L.) can be extracted for several different WIMP masses. Current limits for spin-independent WIMP interactions are depicted in Fig. 19.8, taken from [128]. The Figure includes only a selection of some of the most sensitive experiments. WIMP masses below 4 GeV c−2 are accessible by detectors like CRESST II [124], CDMSlite [115], EDELWEISS [118] and DAMIC [129] because of their low recoil energy thresholds and/or their light absorber nuclei. For WIMP masses above 6 GeV c−2 the best constraints are provided by experiments like XENON1T [130], LUX [131], PANDAX II [132], XENON 100 [133] and Dark Side 50 [134], which are based on massive dual phase (liquid and gas) Xenon or Argon detectors with time projection (TPC) read out. The DAMA experiment has observed

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an annual modulation signal, which they claim is satisfying the requirements of a dark matter annual modulation signal [135]. Their detector is operated in the Gran Sasso Laboratory in Italy (LNGS) and is based on highly radiopure NaI (Tl) crystal scintillators. Similar results, but less significant, were reported by the CoGeNT experiment with a cryogenic Ge detector in the Soudan Underground Laboratory (SUL) [136]. Annual modulations have not been observed by other even more sensitive experiments and the interpretation of a WIMP signal is controversial. In order to better understand the origin of the observed modulation the SABRE experiment is planning to build twin detectors one of which will be placed in the northern hemisphere at the LNGS and the other in the southern hemisphere at the Stanwell Underground Physics Laboratory (SUPL) in Australia [137]. Both detectors will be identical and based on the same target material used in the DAMA experiment. An extraction of the spin-dependent WIMP-nucleon cross section in a model independent way is not possible, since the nuclear and the SUSY degrees of freedom do not decouple from each other. Nevertheless, when using an “odd group” model which assumes that all the nuclear spin is carried by either the protons or the neutrons, whichever are unpaired, WIMP-nucleon cross sections can be deduced. The CDMSlite experiment [115] achieved constraints for spin dependent interactions below WIMP masses of 4 GeV c−2 complementary to LUX, PANDAX, XENON 100 and PICASSO [138]. Several experiments are planning to extend their sensitivity to a wide range of parameter space by operating multi tonnes of target material, reducing the energy thresholds and background level until the irreducible solar, earth and atmospheric neutrino background level is reached, Fig. 19.8. The EURECA (European Underground Rare Event Calorimeter) will bring together researchers from the CRESST

Fig. 19.8 Current limits for spin-independent WIMP interactions are shown

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and EDELWEISS experiments to built a 1 ton cryogenic detector in the Modane Underground Laboratory in France [139]. SUPER CDMS will be operated in the Sudbury Neutrino Observatory (SNOLab) in Canada and is based on cryogenic Ge and Si absorber materials to increase their sensitivity for dark matter interaction cross sections to 10−43 cm2 for masses down to 1 GeV c−2 [140]. The Dark Side 50 collaboration is planning to built a 23 ton dual phase liquid Argon TPC to be operated at the LNGS. The LUX-ZEPLIN (LZ) experiment is currently under construction in the Sanford Laboratory in South Dakota. It uses 10 ton of liquid XENON (dual phase) in a radio-poor double vessel cryostat [142]. The ultimate WIMP detector is proposed by the DARWIN collaboration at the LNGS [141]. It will be based on multi tonnes of liquid Xenon and will fill almost the entire parameter space for spin independent WIMP interaction cross sections down to the background level of neutrino interactions in the detector material as shown in Fig. 19.8. The ambitious project will also be sensitive to other rare interactions like solar axions, galactic axion like particles, neutrinoless double beta decay in 136 Xe and coherent neutrino nucleus scatterings.

19.5.2 Neutrino Mass Studies Since the discovery of neutrino oscillations by the Kamiokande and SuperKamiokande experiments [143] a new chapter in physics started. These findings showed that neutrinos have a mass and that there is new physics to be expected beyond the Standard Model (SM) in particle physics. Among the most pressing questions remain the absolute values of the neutrino masses, since from oscillation experiments only mass differences can be obtained [144], and the Dirac or Majorana type character of the neutrino. The main streams in this field focus upon the search for the neutrinoless double beta decay and the endpoint energy spectrum of beta active nuclei. Cryogenic detectors are particularly well suited for this type of research since they provide excellent energy resolutions, an effective background discrimination and a large choice of candidate nuclei.

19.5.2.1 Neutrinoless Double Beta Decay Double beta decay was first suggested in 1935 by Maria Goeppert Mayer [145]. It is the spontaneous transition from a nucleus (A,Z) to its isobar (A,Z+2). This transition can proceed in two ways: (A,Z)⇒(A,Z+2) + 2 e− + 2 ν¯ e or (A,Z)⇒(A,Z+2) + 2 e− . In the first channel, where two electrons and two antineutrinos are emitted, the lepton number is conserved. It is the second channel, the neutrinoless double beta decay (0νββ), where the lepton number is violated. In this case, with no neutrino in the final state, the energy spectrum of the decay would show in a peak which represents the energy sum of the two electrons. The experimental observation of this process would imply that neutrinos are Majorana particles, meaning that the

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neutrino is not distinguishable from its antiparticle and that it has a non-vanishing 0ν ) one can derive in principle its effective mass. From the measured decay rate (1/T1/2 mass < mν > or a lower limit of it: 0ν (1/T1/2 ) = G0ν (E0 , Z) | M0ν |2 < mν >2

(19.18)

where G0ν (E0 , Z) is an accurately calculable phase space function and M0ν is the nuclear matrix element, which is not very well known [146]. The calculated values of M0ν can vary by factors up to two. Consequently the search for 0νββ should be made with several different nuclei in order to confirm an eventual discovery of this important process. The Milano group has developed an experiment with the name CUORICINO to search for the neutrinoless double beta decay of 130Te. The experiment is located in the Gran Sasso Underground Laboratory. The detector consists of an array of 62 TeO2 crystals with the dimensions 5 × 5 × 5 cm3 (44 crystals) and 3 × 3 × 3 cm3 (18 crystals) and a total mass of 40.7 kg. The crystals are cooled to ∼8 mK and attached to Ge NTD thermistors for phonon detection. Among other possible nuclear candidates (like for example 48 CaF2 , 76 Ge, 100 MoPbO4, 116 CdWO4 , 150 NdF3 , 150 NdGaO ), 130 TeO was chosen because of its high transition energy of 2528.8 ± 3 2 1.3 keV and its large isotopic abundance of 33.8%. Published first results of the CUORICINO experiment [147] show no evidence for the 0νββ decay, but they set 0ν ≥ 1.8 · 1024 yr (90% C.L.) corresponding to a lower limit on the half lifetime T1/2 < mν > ≤ 0.2 to 1.1 eV (depending on nuclear matrix elements). In a next step the collaboration developed CUORE-0 as prototype for a larger detector CUORE. Its basic components consist of 52 TeO2 crystals with dimensions 5 × 5 × 5 cm3 and a total weight of 39 kg corresponding to 10.9 kg 130 Te. CUORE-0 was operated in the CUORICINO cryostat at 12 mK. The data taken from 2013 to 2015 show no evidence for a neutrinoless double beta signal. Combined with the CUORICINO 0ν ≥ 4 · 1024 yr (90% C.L.) corresponding to results a limit on the half lifetime T1/2 < mν > ≤ 270 to 760 meV (depending on nuclear matrix elements) was achieved [148]. CUORE, contains 19 CUORE-0 type towers with 988 TeO2 crystals of a total mass of 741 kg corresponding to 206 kg of 130 Te [149, 150]. The array will be cooled in a large cryostat to 10 mK. It started commissioning early 2017 and aims 0ν ≥ 9 · 1025 yr in 5 years running time [150]. for a sensitivity to reach limits of T1/2 For the future the CUPID collaboration plans to develope a tonne-scale cryogenic detector which will be based on the experience gained with the CUORE experiment [151]. Several experiments investigated other nuclei and set stringent upper limits on the decay rates, for example: KamLand-Zen in 136 Xe [152], EXO-200 in 136 Xe [153], GERDA in76 Ge [154], NEMO-3 in 100 Mo [155]. So far no neutrinoless double beta 0ν ≥ 1.07 · 1026 yr (90% signal was seen. Currently a limit on the half lifetime T1/2 C.L.) corresponding to < mν > ≤ 60 to 165 meV (depending on nuclear matrix elements) was achieved by the KamLand-Zen experiment. An ambitious alternative approach in looking for Majorana versus Dirac type neutrinos is proposed by the

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PTOLEMY experiment in studying the interaction of cosmic relic neutrinos with Tritium [156].

19.5.2.2 Direct Neutrino Mass Measurements So far the best upper limit for the electron neutrino mass of 2.2 eV was obtained from the electron spectroscopy of the tritium decay 3 H ⇒ 3 He + e− + ν¯ e , with a transition energy of 18.6 keV, by the Mainz and the Troitsk experiments [157]. In the near future the KATRIN experiment, which measures the same decay spectrum with a much improved electron spectrometer, will be in operation aiming for a neutrino mass sensitivity down to 0.2 eV [158]. One of the problems with experiments based on a spectroscopic measurement of the emitted electrons is that they yield negative values for the square of the neutrino mass when fitting the electron energy spectrum. This is mainly due to final state interactions (like tritium decays into excited atomic levels of 3 He), which lead to deviations from the expected energy spectrum of the electron. Low temperature calorimeters provide an alternative approach, since they measure the total energy including final state interactions, such as the de-excitation energy of excited atomic levels. However, in order to reach high sensitivity for low neutrino masses the detector has to have an excellent energy resolution and enough counting rate statistics at the beta endpoint energy. The Genoa group [159] pioneered this approach and studied the beta decay of 187 Re ⇒ 187Os + e− + ν¯ e with a cryogenic micro-calorimeter. Their detector was a rhenium single crystal (2 mg) coupled to a Ge NTD thermistor. Rhenium is a super-conductor with a critical temperature of 1.7 K. Natural Re contains 62.8% of 187 Re with an endpoint energy of about 2.6 keV. The operating temperature of the detector was T = 90 mK. In their first attempt they obtained precise values for the beta endpoint energy and the half life of the 187 Re beta decay and were able to obtain an upper limit of the electron neutrino mass of 19 eV (90% CL) or 25 eV (95% CL) [160]. Following this approach the Milan group [161] has built an array of ten thermal detectors for a 187 Re neutrino mass experiment. The detectors were made from AgReO4 crystals with masses between 250 and 350 μg. The crystals were coupled to Si implanted thermistors. Their average energy resolution (FWHM) at the beta endpoint was 28.3 eV, which was constantly monitored by means of fluorescence X-rays. The natural fraction of 187 Re in AgReO yields a decay rate of 5.4 · 10−4 Hz/μg. From a fit to the Curie plot 4 of the 187 Re decay they obtained an upper limit for mν¯ e ≤ 15 eV. Their measured value for the beta endpoint energy is 24653.3 ± 2.1 eV and for the half live is (43.2±0.3)·109 yr. A higher sensitivity to low neutrino masses may be achievable in the future, provided that the energy resolution and the statistics at the beta endpoint energy can be improved significantly. The latter may raise a problem for thermal phonon detectors, since their signals are rather slow and therefore limit the counting rate capability to several Hz. With their Rhenium cryogenic micro-calorimeters the Genoa group [162, 163] and the Milano collaboration [164] were also able to measure interactions between

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the emitted beta particle and its local environment, known as beta environmental fine structure (BEFS). The BESF signal originates from the interference of the outgoing beta electron wave and the reflected wave from the atoms in the neighbourhood. BEFS is similar to the well known Extended X-ray Absorption Fine Structure (EXAFS) method. Their results demonstrated that cryogenic micro-calorimeters may also offer complementary new ways for material sciences to study molecular and crystalline structures. Currently several groups MARE [165, 166], ECHo [167], HOLMES [168], NUMECS [169], are investigating the possibility to measure the electron neutrino mass from the Electron Capture (EC) decay spectrum of Holmium (163 Ho) using cryogenic micro-calorimeters. This approach was originally suggested by A. de Rujula and M. Lusignoli in 1982 [170]. 163 Ho decays via EC into 163 Dy with a half life of 4570 years and a decay energy of 2.833 keV. It does not occur naturally and it is not commercially available. It has to be produced by neutron or proton irradiation. After purification the 163 Ho atoms have to be implanted into a suitable absorber material of the micro-calorimeter. In order to reach a neutrino mass sensitivity in the sub-eV region a total 163Ho activity of several MBq is required. Since the activity of a single micro-calorimeter should not exceed 100 Bq the total 163 Ho activity has to be distributed over a large number of pixels ( 105 ). The groups are devoting much effort in developing micro-calorimeters with energy and time resolutions of the order of 1 eV and 1μs respectively. Various thermal sensors, like TES, MMC and MKID, are considered. Multiplexing schemes have still to be invented to be able to read out the enormous number of pixels.  As already mentioned above, an upper limit for all neutrino masses of mν ≤ 0.234 eV c−2 was derived from cosmology. It will still take some efforts to reach or go below these limits in the near future with direct mass measurements. A review of direct neutrino mass searches can be found in [171].

19.5.3 Astrophysics Modern astrophysics addresses a large list of topics: Formation of galaxies and galaxy clusters, the composition of the intergalactic medium, formation and evolution of black holes and their role in galaxy formation, matter under extreme conditions (matter in gravitational fields near black holes, matter inside neutron stars), supernovae remnants, accretion powered systems with white dwarfs, interstellar plasmas and cosmic microwave background radiation (CMB). The investigation of these topics requires optical instruments with broad band capability, high spectral resolving power, efficient photon counting and large area imaging properties. The radiation received from astrophysical objects spans from microwaves, in the case of CMB, to high energy gamma rays. The subjects discussed here can be divided into three categories, X-ray, optical/ultraviolet (O/UV) and CMB observations. In order to avoid the absorptive power of the earth atmosphere many of the instruments are operated in orbiting observatories, in sounding rockets or in balloons. Progress in

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this rapidly growing field of science is constantly asking for new instrumentation and new technologies. Cryogenic detectors are playing a key role in these developments providing very broad-band, imaging spectrometers with high resolving power. They also feature high quantum efficiencies, single photon detection and timing capabilities. The observation of large-scale objects, however, needs spatialspectral imaging devices with a wide field of view requiring cryogenic detectors to be produced in large pixel arrays. The fabrication and the readout of these arrays remains still a big challenge.

19.5.3.1 X-Ray Astrophysics The orbital X-ray observatories Chandra and XMM-Newton contained CCD cameras for large field imaging and dispersive spectrometers for narrow field high spectral resolution. Cryogenic devices are able to combine both features in one instrument. Although they have not yet reached the imaging potential of the 2.5 megapixel CCD camera on XMM-Newton and the resolving power E/EF W H M = 1000 at E = 1 keV of the dispersive spectrometer on Chandra, their capabilities are in many ways complementary. For example, the resolving power of cryogenic devices increases with increasing energy and is above 2 keV better than the resolving power of dispersive and grating spectrometers, which decreases with increasing energy. Since the cryogenic pixel array provides a complete spectral image of the source at the focal plane its resolving power is independent of the source size. Cryogenic detectors also provide precise timing information for each photon allowing to observe rapidly varying sources such as pulsars, etc. They cover a wide range of photon energies (0.05–10 keV) with a quantum efficiency of nearly 100%, which is 5 times better than the quantum efficiency (20%) of CCDs. The first space-borne cryogenic X-ray Quantum Calorimeter (XQC), a collaboration between the Universities of Wiskonsin, Maryland and the NASA Goddard Space Flight Center, was flown three times on a sounding rocket starting in 1995 [172]. The rockets achieved an altitude of 240 km providing 240 s observation above 165 km per flight. The XQC was equipped with a 2 × 18 micro-calorimeter array consisting of HgTe X-ray absorbers and doped silicon thermistors yielding an energy resolution of 9 eV across the spectral band. The pixel size was 1 mm2 . The micro-calorimeters were operated at 60 mK. It is interesting to note that, when recovering the payload after each flight, the dewar still contained some liquid helium. The purpose of the mission was to study the soft X-ray emission in the band of 0.03–1 keV. The physics of the diffuse interstellar X-ray emission is not very well understood. It seems that a large component is due to collisional excitations of particles in an interstellar gas with temperatures of a few 106 K. A detailed spectral analysis would allow to determine the physical state and the composition of the gas. Since the interstellar gas occupies a large fraction of the volume within the galactic disk, it plays a major role in the formation of stars and the evolution of the galaxy. The results of this experiment and their implications are discussed in [172]. As a next step the Japan-USA collaboration has put an X-ray spectrometer

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(XRS) on board of the Astro-E2 X-ray Suzaku satellite which was launched in July 2005. This instrument is equipped with 32 pixels of micro-calorimeters (HgTe) and semiconducting thermistors and is an improvement over the XQC spectrometer in terms of fabrication techniques, thermal noise, energy resolution of 7 eV across the operating band of 0.03–10 keV and observation time [173, 174]. The pixels are 0.624 mm2 and arranged in a 6 × 6 array giving a field of view of 2.9 × 2.9 arcmin. The observatory is looking at the interstellar medium in our and neighboring galaxies as well as at supernovae remnants. The investigations include super massive black holes and the clocking of their spin rate. The next step is to develop cryogenic detectors with increased pixel numbers (30 × 30) and energy resolutions of 2 eV, which would be able to replace dispersive spectrometers in future experiments. Superconducting micro-calorimeters with TES sensors have the potential to reach energy resolutions of 2 eV and are likely to replace semiconducting thermistors. One of the problems with cryogenic detectors is that they do not scale as well as CCDs, which are able to clock the charges from the center of the arrays to the edge using a serial read out. Cryogenic detectors rely on individual readout of each pixel. Another problem is the power dissipation in large arrays. To solve some of these problems, dissipation-free metallic magnetic calorimeters (MMC) and microwave kinetic inductance detectors (MKID) [80] are among the considered possibilities. Details of these new developments can be found in the proceedings of LTD. Large cryogenic detector arrays are planned for ATHENA (Advanced Telescope for High Energy Astrophysics), a future X-ray telescope of the European Space Agency. It is designed to investigate the formation and evolution of large scale galaxy clusters and the formation and grows of super massive black holes. The launch of ATHENA is planned for 2028.

19.5.3.2 Optical/UV and CMB Astrophysics Since the first optical photon detection with STJ’s in 1993 [175] and TES’s in 1998 [176] a new detection concept was introduced in the field of Optical/UV astrophysics. Further developments demonstrated the potential of these single photon detection devices to combine spectral resolution, time resolution and imaging in a broad frequency band (near infrared to ultraviolet) with high quantum efficiency. The principle of these spectrophotometers is based on the fact that for a superconductor with a gap energy of typically 1 meV an optical photon of 1 eV represents a large amount of energy. Thus a photon impinging on a superconductor like for example Ta creates a large amount of quasi-particles leading to measurable tunnel current across a voltage biased junction. A first cryogenic camera S-Cam1 with a 6 × 6 array of Ta STJs (with a pixel size of 25 × 25 μm2 ) was developed by ESTEC/ESA and 1999 installed in the 4 m William Herschel telescope on La Palma (Spain) [75]. For a first proof of principle of this new technique the telescope was directed towards the Crab pulsar with an already known periodicity of 33 ms. The photon timing information was recorded with a 5 μs accuracy with respect to the GPS timing signals. Following the success of the demonstrator model S-Cam1

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and of the improved model S-Cam2 a new camera S-Cam3 consisting of a 10 × 12 Ta STJ pixel array (with pixel dimension 33 × 33 μm2 ) was installed at the ESA 1m Optical Ground Station Telescope in Tenerife (Spain). The STJ structure is 100 nm Ta/30 nm Al//AlOx//30 nm Al/100 nm Ta. The camera covers a wavelength range of 340–740 nm with a wavelength resolution of 35 nm at λ = 500 nm and has a pulse decay time of 21 μs [177]. The Stanford-NIST (National Institute of Standards and Technology) collaboration has also developed a camera with an 8 × 8 pixel array based on tungsten TES sensors on a Si substrate [178]. Each pixel has a sensitive area of 24 × 24 μm2 and the array has a 36 × 36 μm center to center spacing. In order to improve the array fill factor a reflection mask is positioned over the inter-pixel gaps. For both STJ and TES spectro-photometers thermal infrared (IR) background radiation, which increases rapidly with wavelength above 2 μm, is of concern. Special IR blocking filters have to be employed in order to extend the wavelengths range of photons from 0.3 μm out to 1.7 μm. A 4 pixel prototype of the Stanford-NIST TES instrument was already mounted at the 2.7 m Smith Telescope at McDonald Observatory (U.S.A) and observed a number of sources including spin powered pulsars and accreting white dwarf systems. The Crab pulsar served as a source to calibrate and tune the system. The obtained data are published in [179]. Already these first results have shown that STJ or TES based spectrophotometers are in principal very promising instruments to study fast time variable sources like pulsars and black hole binaries as well as faint objects, like galaxies in their state of formation. However, in order to extend the observations from point sources to extended objects much larger pixel arrays are required. Future developments concentrate on a suitable multiplexing system in order to increase the number of pixels, which are presently limited by the wiring on the chip and the size of the readout electronics. SQUID multiplexing readout systems [180– 182] as well as Distributed Read-out Imaging Devices (DROID) [74, 183], in which a single absorber strip is connected to two separate STJs on either side to provide imaging capabilties from the ratio of the two signal pulses, are under study. However, these devices are slower than small pixel devices and can handle only lower count rates. A much faster device is the superconducting MKID detector [80], which allows a simple frequency-domain approach to multiplexing and profits from the rapid advances in wireless communication electronics. A more detailed review can be found in [184, 185]. A camera, ACRONS, for optical and near infrared spectroscopy has been developed. The camera contains a 2024 pixel array of cryogenic MIKD detectors. The device is able to detect individual photons with a time resolution of 2 μs and simultaneous energy information [80]. The instrument has been used for optical observations of the Crab pulsar [186]. Large cryogenic pixel antennas have been developed for ground-based and spaceborne CMB polarization measurements. These devices aim to be sensitive to the detection of the so-called primordial E- and B-modes, which would appear as curling patterns in the polarization measurements. E-modes arise from the density perturbations while B- modes are created by gravitational waves in the early universe. The two modes are distinguishable through their characteristic patterns. However, B-mode signals are expected to be an order of magnitude weaker than E-

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modes. Nevertheless, B-modes are of particular interest since they would provide revealing insight into the inflationary scenario of the early universe signaling the effect of primordial gravity waves. There are several instruments, which use TES based cryogenic bolometers, in operation: EBEX [187] and SPIDER [188] are balloon-borne experiments, overflying the Antarktis. POLARBEAR [189] is an instrument, which is coupled to the HUAN TRAN Teleskope at the James Ax Observatory in Chile. BICEP2 and the Keck Array [190] are located at the Amundsen-Scott South Pole Station. The PLANCK Satellite [191] carried 48 cryogenic bolometers operating at 100 mK in outer space. In March 2014 the BICEP2 collaboration reported the detection of B-modes [190]. However the measurement was received with some skepticism and David Spergel argued that the observation could be the result of light scatterings off dust in our galaxy. In September 2014 the PLANCK team [191] concluded that their very accurate measurement of the dust is consistent with the signal reported by BICEP2. In 2015 a joint analysis of BICEP2 and PLANCK was published concluding that the signal could be entirely attributed to the dust in our galaxy [192].

19.6 Applications Electron probe X-ray microanalysis (EPMA) is one of the most powerful methods applied in material sciences. It is based on the excitation of characteristic Xrays of target materials by high current electron beams in the energy range of several keV. EPMA finds its application in the analysis of contaminant particles and defects in semiconductor device production as well as in the failure analysis of mechanical parts. It is often used in the X-ray analysis of chemical shifts which are caused by changes in the electron binding due to chemical bonding as well as in many other sciences (material, geology, biology and ecology). The Xrays are conventionally measured by semiconducting detectors (Si-EDS), which are used as energy dispersive spectrometers covering a wide range of the X-ray spectrum, and/or by wavelength dispersive spectrometers (WDS). Both devices have complementary features. WDS, based on Bragg diffraction spectrometry, has a typical energy resolution of 2–15 eV (FWHM) over a large X-ray range. However, the diffraction limits the bandwidth of the X-rays through the spectrometer as well as the target size, which acts as a point source, and makes serial measurements necessary, which is rather time consuming. Contrary, the Si-EDS measures the entire X-ray spectrum from every location of the target simultaneously, but with a typical energy resolution of 130 eV. Si-EDS is therefore optimally suited for a quick but more qualitative analysis. Cryogenic micro-calorimeter EDS provides the ideal combination of the high resolution WDS and the broadband features of the energy dispersive EDS. The application of cryogenic detectors for EPMA was first introduced by Lesyna et al. [193]. The NIST group developed a prototype TES based micro-calorimeter which is suitable for industrial applications [42, 194, 195]. It consists of a Bi absorber and a Al-Ag or Cu-Mo bi-layer TES sensor. It covers

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an area of 0.4 × 0.4 mm2 and yields an energy resolution of 2 eV at 1.5 keV and 4.5 eV at 6 keV. The detector is cooled to 100 mK by a compact adiabatic demagnetization refrigerator and is mounted on a scanning electron microscope. Cryogenic refrigerators for this and various other applications are commercially available [196]. In spite of its excellent resolving power (see Fig. 19.2) the microcalorimeter EDS has still two shortcomings. It has a limited counting rate capability (1 kHz compared to 100 kHz WDS and 25 kHz Si-EDS) and a small effective detector surface. To compensate for the latter the NIST group developed an Xray focusing device using poly-capillary optics. The device consists of many fused tapered glass capillaries which focus the X-rays by means of internal reflection onto the micro-calorimeter increasing its effective area. Another solution under study is a multiplexed micro-calorimeter array with a possible loss in resolution due to the variability of individual detectors [182]. Nevertheless, TES based cryogenic microcalorimeters EDS have already demonstrated major advances of EPMA in scientific and industrial applications. Time of flight mass spectrometry (ToF-MS) of biological molecules using cryogenic detectors was first introduced by D. Twerenbold [197]. The main advantage of cryogenic calorimeters over traditionally employed micro-channel plates (MP) is that the former are recording the total kinetic energy of an accelerated molecule with high efficiency, independent of its mass, while the efficiency of the latter is decreasing with increasing mass due to the reduction of the ionization signal. ToFMS equipped with MP lose rapidly in sensitivity for masses above 20 kDa (proton masses). The disadvantages of cryogenic detectors are: first, they cover only a rather small area of ∼1 mm2 while a MP with 4 cm2 will cover most of the beam spot size of a spectrometer; second, the timing signals of the cryogenic calorimeters are in the range of μs and therefore much slower than ns signals from MP, which degrades the flight time measurements and thus the accuracy of the molecular mass measurements. A good review of early developments can be found in [198]. After the early prototype experiences made with STJs [199, 200] and NIS (Normalconductor/Isolator/Super-conductor) tunnel junctions [201], which provided only very small impact areas, super-conducting phase transition thermometers (SPT) with better time of flight resolutions and larger impact areas of 3 × 3 mm2 were developed [202–204]. These devices consist of thin super-conducting Nb meanders, or super-conducting films in thermal contact with an absorber, which are current biased and locally driven to normal conducting upon impact of an ion. A voltage amplifier is used to measure the signal pulse. Cryogenic detectors as high resolution γ -ray, α and neutron spectrometers also found applications in nuclear material analysis, as broad band micro-calorimeters in Electron Beam Ion Traps (EBITs) and in synchrotrons for fluorescence-detected X-ray absorption spectroscopy (XAS) [202]. They are also employed in nuclear and heavy ion physics [205].

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19.7 Summary Cryogenic detectors have been developed to explore new frontiers in astro and particle physics. Their main advantages over more conventional devices are their superior energy resolution and their sensitivity to very low energy transfers. However, most thermal detectors operate at mK temperatures requiring complex refrigeration systems and they have limited counting rate capabilities (1 Hz– 1 kHz). Today the most frequently used thermometers for calorimeters operating in near equilibrium mode are doped semiconductors (thermistors), superconducting transition edge sensors (TES) and metallic paramagnets (MMC). Because they are easy to handle and commercially available thermistors are quite popular. They have, however, the disadvantage of having to deal with Joule heating introduced by their readout circuit. The most advanced technology is provided by the TES sensors in connection with an auto-biasing electrothermal feedback system. This system reduces the effect of Joule heating, stabilizes the operating temperature and is self-calibrating, which turns out to be advantageous also for the operation of large detector arrays. The main advantage of MMCs is their magnetic inductive readout, which does not dissipate power into the system. This feature makes MMC attractive for applications, where large detector arrays are required. Non-equilibrium detectors like superconducting tunnel junctions (STJ), superheated superconducting granules (SSG) and microwave kinetic inductance devices (MKID) are based on the production and detection of quasi-particles as a result of Cooper pair breaking in the superconductor. These devices are intrinsically faster, providing higher rate capabilities (10 kHz and more) and good timing properties suitable for coincident measurements with external detectors. Because of their sensitivity to low energy photons arrays of STJs are frequently employed in infrared and optical telescopes, but also efficiently used in x-ray spectroscopy. SSG detectors with inductive readout have the potential to reach very low energy thresholds (order of several eV) which would be advantageous for various applications like for example in neutrino physics (coherent neutrino scattering, etc.). But the practical realization of SSG detectors is still very challenging. MKIDs provide an elegant way to readout large detector arrays by coupling an array of many resonators with slightly different resonance frequencies to a common transmission line with a single signal amplifier. Due to this feature they are very suited for the future instrumentation of astrophysical observatories and other applications. Cryogenic calorimeters with a large detector mass for dark matter searches and neutrino physics as well as large detector arrays for astrophysical measurements and other practical applications are under intense developments. Despite the enormous progress made in the past their fabrication and readout remain still a challenge.

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Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made. The images or other third party material in this chapter are included in the chapter’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Chapter 20

Detectors in Medicine and Biology P. Lecoq

20.1 Dosimetry and Medical Imaging The invention by Crookes at the end of the nineteenth century of a device called spinthariscope, which made use of the scintillating properties of Lead Sulfide allowed Rutherford to count α particles in an experiment, opening the way towards modern dosimetry. When at the same time Wilhelm C. Roentgen, also using a similar device, was able to record the first X-ray picture of his wife’s hand 2 weeks only after the X-ray discovery, he initiated the first and fastest technology transfer between particle physics and medical imaging and the beginning of a long and common history. Since that time, physics, and particularly particle physics has contributed to a significant amount to the development of instrumentation for research, diagnosis and therapy in the biomedical area. This has been a direct consequence, one century ago, of the recognition of the role of ionizing radiation for medical imaging as well as for therapy.

20.1.1 Radiotherapy and Dosimetry The curative role of ionizing radiation for the treatment of skin cancers has been exploited in the beginning of the twentieth century through the pioneering work of some physicists and medical doctors in France and in Sweden. This activity has very much progressed with the spectacular developments in the field of accelerators,

P. Lecoq () CERN, Geneva, Switzerland e-mail: [email protected] © The Author(s) 2020 C. W. Fabjan, H. Schopper (eds.), Particle Physics Reference Library, https://doi.org/10.1007/978-3-030-35318-6_20

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beam control and radioisotope production. Today radiotherapy is an essential modality in the overall treatment of cancer for about 40% of all patients treated. Conventional radiotherapy (RT) with X-rays and electrons is used to treat around 20,000 patients per 10 million inhabitants each year. The main aim of radiation therapy is to deliver a maximally effective dose of radiation to a designated tumour site while sparing the surrounding healthy tissues as much as possible. The most common approach, also called teletherapy, consists in bombarding the tumour tissue with ionizing radiation from the outside of the patient’s body. Depending on the depth of the tumour, soft or hard X-rays or more penetrating γ-rays produced by a 60 Co source or by a linac electron accelerator are used. However, conventional X-ray or γ-ray radiation therapy is characterized by almost exponential attenuation and absorption, and consequently delivers the maximum energy near the beam entrance. It also continues to deposit significant energy at distances beyond the cancer target. To compensate for the disadvantageous depth-dose characteristics of X-rays and γ-rays and to better conform the radiation dose distribution to the shape of the tumour, the radiation oncologists use complex Conformal and Intensity Modulated techniques (IMRT) [1]. The patient is irradiated from different angles, the intensity of the source and the aperture of the collimators being optimized by a computer controlled irradiation plan in order to shape the tumour radiation field as precisely as possible. Another way to spare as much as possible healthy tissues is to use short range ionizing radiation such as β or α particles produced by the decay of unstable isotopes directly injected into the tumour. This method, called brachytherapy, has been originally developed for the thyroid cancer with the injection of 153 I directly into the nodules of the thyroid gland. It is also used in other small organs such as prostate or saliva gland cancer. Following these trends a new generation of minimally invasive surgical tools appears in hospitals, allowing to precisely access deep tumours from the exterior of the body (gamma-knife), or by using brachytherapy techniques, i.e. by injecting radioisotopes directly in the tumour (beta-knife, alpha-knife and perhaps soon Auger-knife). More recently the radio-immunotherapy method has been successfully developed: instead of being directly inserted in the tumour, the radioactive isotopes can be attached by bioengineering techniques to a selective vector, which will bind to specific antibody receptors on the membranes of the cells to be destroyed. Typical examples are the use of the α emitter 213 Bi for the treatment of leukaemia and of the β-emitter 90 Y for the treatment of glioblastoma. In 1946 Robert Wilson, physicist and founder of Fermilab, proposed the use of hadron beams for cancer treatment. This idea was first applied at the Lawrence Berkeley Laboratory (LBL) where 30 patients were treated with protons between 1954–1957. Hadrontherapy is now a field in rapid progress with a number of ambitious projects in Europe, Japan and USA [2], exploiting the attractive property of protons and even more of light ions like carbon to release the major part of their kinetic energy in the so-called Bragg peak at the end of their range in matter (Fig. 20.1).

Absorbed Relative Dose [%]

20 Detectors in Medicine and Biology

100 80 60 40 20 0

915

Bragg peak Linac Cobalt 60 Proton Tail

0 Body surface

Light ion 5 10 (carbon) 15 Depth from body surface [cm] Tumor site

Fig. 20.1 Bragg peak for an ion beam in the brain of a patient. The insert shows the energy absorbed by tissues as a function of depth for different radiation sources (Courtesy U. Amaldi)

It is particularly important to treat the disease with the minimum harm for surrounding healthy tissues. In the last centimeters of their range the Linear Energy Transfer (LET) of protons or even more of carbon ions is much larger than the one of X-rays (low-LET radiations). The resulting DNA damages include more complex double strand breaks and lethal chromosomal aberrations, which cannot be repaired by the normal cellular mechanisms. The effects produced at the end of the range are therefore qualitatively different from those produced by X- or γ-rays and open the way to a strategy to overcome radio-resistance, often due to hypoxia of the tumour cells. For these reasons carbon ions with their higher relative biological effectiveness (RBE) at the end of their range, of around a factor of three higher than X-rays, can treat tumours that are normally resistant to X-rays and possibly protons. This treatment is particularly applicable to deep tumours in the brain or in the neck as well as ocular melanoma. Whatever the detailed modality of the treatment planning, precise dosimetry is mandatory to develop an optimal arrangement of radiation portals to spare normal and radiosensitive tissues while applying a prescribed dose to the targeted disease volume. This involves the use of computerized treatment plan optimization tools achieving a better dose conformity and minimizing the total energy deposition to the normal tissues (Fig. 20.2). It requires a precise determination and simulation of the attenuation coefficients in the different tissues along the beam. These data are obtained from high performance anatomic imaging modalities such as X-ray computed tomography (CT) and magnetic resonance (MRI).

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130 117 103 90 77 63 50 37 23 10

X ray beam

Dose level

Dose % 108 95 80 70 60 50 40

Target

30

Fig. 20.2 Dosimetry for a brain tumour in the case of one (left) or nine crossed (right) X-ray beams. The treatment plan is based on a tumour irradiation of at least 90 Gy (Courtesy U. Amaldi)

For the particular case of hadrontherapy on-line dosimetry in the tissues is in principle possible. It relies on the production of positron emitter isotopes produced by beam spallation (10 C and 11 C for 12C beam) or target fragmentation during the irradiation treatment. The two 511 keV γ produced by the positron annihilation can be detected by an in-line positron emission tomography (PET) to precisely and quantitatively map the absorbed dose in the tumour and surrounding tissues. Although challenging because of the timing and high sensitivity requirements this approach is very promising and a number of groups are working on it worldwide [3].

20.1.2 Status of Medical Imaging The field of medical imaging is in rapid evolution and is based on five different modalities: X-ray radiology (standard, digital and CT), isotopic imaging (positron emission tomography, PET, and single photon emission computed tomography SPECT), ultrasound (absorption, Doppler), magnetic resonance (MRI, spectroscopy, functional), and electrophysiology with electro- and magnetoencephalography (EEG and MEG). More recently, direct optical techniques like bioluminescence and infrared transmission are also emerging as powerful imaging tools for non-too-deep organs. For a long time imaging has been anatomical and restricted to the visualization of the structure and morphology of tissues allowing the determination of morphometric parameters. With the advent of nuclear imaging modalities (PET and SPECT) and of the blood oxygen level dependent (BOLD) technique in magnetic resonance imaging (MRI) functional imaging became possible and medical doctors can now see organs at work. Functional parameters are now accessible in vivo and in real

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time, such as vascular permeability, haemodynamics, tissue oxygenation or hypoxia, central nervous system activity, metabolites activity, just to cite a few. In the current clinical practice medical imaging is aiming at the in-vivo anatomic and functional visualization of organs in a non- or minimally invasive way. Isotopic imaging, in particular PET, currently enjoys a spectacular development. Isotopic imaging consists in injecting into a patient a molecule involved in a specific metabolic function so that this molecule will preferentially be fixed on the organs or tumours where the function is at work. The molecule has been labeled beforehand with a radioisotope emitting gamma photons (Single Photon Emission Computed Tomography or SPECT) or with a positron emitting isotope (Positron Emission Tomography or PET). In the latter case, the positron annihilates very quickly on contact with ordinary matter, emitting two gamma photons located on the same axis called the line of response (LOR) but in opposite directions with a precise energy of 511 keV each. Analyzing enough of these gamma photons, either single for SPECT or in pairs for PET, makes it possible to reconstruct the image of the area (organ, tumour) where the tracer focused. Since the beginning to the twenty-first century a new generation of machines became available, which combine anatomic and functional features: the PET/CT. This dual modality system allows the superposition of the high sensitivity functional image from the PET on the precise anatomic picture of the CT scanner. PET/CT has now become a standard in the majority of hospitals, particularly for oncology. This trends for multimodal imaging systems is increasing both for clinical and for research applications (Fig. 20.3).

Fig. 20.3 Abdominal slice of a 78 year-old male, with biopsy-proven prostate adenocarcinoma and penile adenocarcinoma. Focal uptake in the prostate bed and in the penile shaft (full arrows). Multiple foci in the pelvis compatible with skeletal metastases (dashed arrows) (Courtesy D. Townsend)

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Table 20.1 Comparison of the performances of four imaging modalities Imaging modality PET

SPECT MRI

CT

Type of imaging Functional and molecular (picomolar sensitivity) Functional and molecular Anatomical Functional (millimolar sensitivity) Anatomical

Examination time 10–20 (whole body)

Spatial resolution 3–5mm

10–20 (per 40 cm filed)

6–8 mm

30–60

0.5 mm

2026) (7,7 TeV) FCC p-p (50, 50 TeV) foreseen >2040

σtot [barn] 4n

0 L [cm−2 s−1 ] ∼ φdt [neq cm−2 ] Dose in Si [Gy]

5.0 · 1035

3p few 1034 3p 10−3 7 · 1031 2 (Q < 100 GeV) 70 m 1.7 · 1032 100 m 100 m

0

Q i . Q hit < 0

charge induced in the neighbors

hit

pad hit

Fig. 21.11 Explanation of trapping induced charge sharing

As Uw depends on the geometry only it is obvious that it is possible to optimize the electrode design for maximum signal. However, as the paramount parameter for any detector is its signal-over-noise ratio, the optimization should also include the inter-strip capacitance and leakage current of electrode both affecting the noise. Charge collection in segmented devices leads to “signal cross-talk” as described in the section on Signal processing 6.2.1. The bi-polar current pulses in the neighboring electrodes (see e.g. Uw in Fig. 21.10) yield zero net charge for integration time larger than the drift time (see Signal processing Fig. 6.2). In irradiated detector some of the carriers are trapped and do not complete their drift. Therefore the integrals of the bipolar pulses do not vanish. A significant amount of charge can appear in the neighbors adding to the usual charge shared by diffusion (see explanation in Fig. 21.11). Unlike diffusion, where the polarity of the induced charge is equal for all electrodes, the trapping can result in charges of both polarities. If electrodes collect carriers with smaller μ · τeff , the polarity of the charge is the same for all electrodes. Otherwise the polarity of the charge induced in the neighbors is of opposite sign compared to the hit electrode [27, 28] The effect can be used to enhance the spatial resolution due to larger charge sharing at the expense of smaller charge collection efficiency or vice versa.

21 Solid State Detectors for High Radiation Environments

(a)

(b) Al

SiO2

Al 2

p+ W

987

n+

n bulk

n,p bulk

n+

p+

p−stop

p−spray

Fig. 21.12 Schematic view of (a) p+ − n − n+ and (b) n+ − n − p+ , n+ − p − p+ strip detectors (AC coupled)

21.4.2 Silicon Detectors Silicon is by far the most widely used semiconductor detector material. A large majority of silicon particle detectors exploit the asymmetric p − n junction bias in the reverse mode as a basic element. Up to recently the detector grade silicon was produced by the so called float zone (FZ) technique, where concentration of impurities and dopants can be precisely controlled to very low values (∼1011 cm−3 ). The step further in radiation hardening of silicon detectors was the enrichment of the float zone silicon through oxygen diffusion (DOFZ). Recently, detectors were processed on Czochralski1 and epitaxially grown silicon and are in some respects radiation harder than float zone detectors. Most of the detectors used up to now were made on n-type silicon with p+ readout electrodes (see Fig. 21.12a), which collect holes. Electrons have larger μ τeff in silicon, hence n+ readout electrodes are more appropriate for high radiation environments where the loss of charge collection efficiency is the major problem. They are mostly realized by segmentation of n+ side of the n-type bulk (see Fig. 21.12b), which however requires more complex processing on both detector sides. The double sided processing can be avoided by using p-type bulk material with n+ electrodes [29]. This is the preferred choice silicon detector type at HL-LHC.

21.4.2.1 Effective Doping Concentration The defects produced by irradiation lead to change of the effective doping concentration. The main radiation induced defects responsible for the change of effective dopant concentration can be found in Fig. 21.6 and consist of both donors and acceptors.

1 If magnetic field is used to control the melt flow in crucible the process is called MagneticCzochralski.

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Fig. 21.13 (a) Effective doping concentration in standard silicon, measured immediately after neutron irradiation [30] (b) Evolution of Neff evolution with time after irradiation [31]

It is a well established, that irradiation by any particle introduces effectively negative space charge in detectors processed on float zone silicon, which is most commonly used. The change in effective doping concentration is reflected in the full depletion voltage Vf d , needed to establish the electric field in the entire detector: Vf d =

e0 |Neff | W 2 . 20 

(21.30)

The Vf d of initially n-type detectors (p+ −n−n+ ), therefore decreases to the point, where the negative space charge prevails, so called space charge sign inversion point (SCSI). The Neff turns to negative and depleted region grows from the n+ contact at the back. The Vf d thereafter continues to increase with fluence beyond any tolerable value, which is usually set by the breakdown of a device (see Fig. 21.13a). The space charge of p-type detectors (n+ − p − p+ ) remains negative with irradiation so that the main junction stays always at the front n+ − p contact. For both detector types not only deep radiation induced defects are created, but also initial shallow dopants are electrically deactivated (removed)—so called initial dopant removal. The initial dopant removal impacts to large extent the performance of some detector technologies such as Low Gain Avalanche Detectors and depleted CMOS detectors, which will be reviewed later.

Evolution of Effective Dopant Concentration—Hamburg Model After the irradiation the defects responsible for space charge evolve with time according to defect dynamics described by Eqs. (21.3), (21.4). The time scale of these processes varies from days to years already at close to room temperatures which makes the annealing studies lengthy procedures. At elevated temperature the underlying defect kinetics can be accelerated, and thus the simulation of the damage investigation at real experiments spanning several years is possible in weeks.

21 Solid State Detectors for High Radiation Environments

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The radiation induced change in the effective doping concentration is due to historical reasons defined as Neff = Neff,0 − Neff (t), where Neff,0 denotes the initial doping concentration. The fact that the radiation introduced space charge is negative means that Neff is positive. The evolution of Neff after irradiation is shown in Fig. 21.13b. Neff initially decreases, reaches its minimum and then starts to increase. The measured evolution can be described by a so called Hamburg model, which assumes three defects [31] all of them obeying first order kinetics (see Eq. (21.3)). The initial decrease of Neff is associated with decay of effective acceptors (Na ). After a few days at room temperature a plateau, determined by defects stable in time (Nc ), is reached. At late stages of annealing effective acceptors are formed again (NY ) over approximately a year at room temperature. The corresponding equations are: Neff = Neff,0 − Neff = Na (, t) + Nc + Ny (, t) (21.31)     t t Neff = ga eq exp − + Nc + gY eq (1 − exp − ) (21.32) τa τra   (21.33) Nc = ±Nid (1 − η(1 − exp −c · eq )) + gc eq , where ga , gc and gY describe the introduction rates of defects responsible for the corresponding part of the damage and τa and τra the time constants of initial and late stages of annealing. The stable part of the damage incorporates also initial dopant removal, where ±Nid (negative/positive sign for donors/acceptors) denotes the concentration of initial dopants, η fraction of removed dopants and c the removal constant. Displacement of the initial dopant from the lattice site, deactivates it. Once in the interstitial position, initial dopants (mainly boron and phosphorous) can react with other defects leading to possibly new electrically active defects. The new defects formed can also be charged, hence the removal can be partial, i.e. Nid = Neff,0 [32, 33]. For example, the interstitial boron can undergo different reactions with impurities forming both donor and acceptor like defects [32]. As the reactions can take place also with impurities the removal rate depends on their concentration. The initial donor (phosphorous) removal was intensively studied for high resistivity p+ − n − n+ detectors [34], where initial donor removal is attributed to formation of electrically inactive Vacancy-Phosphorous (V-P) complex. The rate of removal was found to depend on initial concentration with Nid × c ≈ 0.008 cm−1 . The reason for such relation is unclear. It was observed that donor removal is complete for charge hadron irradiated detectors while around half of the initial donors remain effectively active after neutron irradiations (η ∼ 0.45 − 0.7). The initial acceptor (boron) removal was much less studied in the n+ − p − p+ particle detectors, more for solar cells [35]. The required radiation hardness of ptype detectors for HL-LHC is such that deep acceptors exceed the concentration of

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Fig. 21.14 Initial acceptor removal rate dependence on initial dopant concentration. The data were obtained from measurements with different detectors/technology: pad diodes (float zone and epitaxial), depleted (HV) CMOS and LGADs. The red markers show neutron irradiations and the blue markers show fast charged hadron irradiations. The red and blue arrows guide the eye. Data from Refs. [36–41]

Table 21.4 The survey of Hamburg model parameters for standard and diffusion oxygenated float zone detectors

ga [cm−1 ] τa [h at 20 ◦ C] gc [cm−1 ] gY [cm−1 ] τra [days at 20 ◦ C]

Standard FZ Neutrons 0.018 55 0.015 0.052 480

Charged hadrons – – 0.019 0.066 500

Diffusion Oxygenated FZ Neutrons Charged hadrons 0.014 – 70 – 0.02 0.0053 0.048 0.028 800 950

The uncertainty in the parameters is of order 10% and mainly comes from variation of silicon materials used

initial ones by far, hence their removal was not in focus. However, new detector technologies (LGAD, depleted CMOS) with significant/dominant concentration of initial dopants also after foreseen fluences, triggered extensive studies of initial acceptor removal. Similarly to donor removal c was found to depend on initial concentration as shown in Fig. 21.14. The rate of removal is around two times larger for fast charged hadrons and only for large initial dopant concentrations the removal is complete (η ≈ 1). The parameters of the Hamburg model related to radiation induced defects (deep traps) are given in the Table 21.4 and are valid for p- and n-type silicon detectors. For reasons that will be explained later, the model parameters are also shown for FZ detectors which were deliberately enriched by oxygen.

21 Solid State Detectors for High Radiation Environments

991

The time constants of initial (τa ) and late stage annealing (τra ) can be scaled to different annealing temperatures by using Eq. (21.9). The activation energies for initial and long term annealing are Era ≈ 1.31 eV and Ea ≈ 1.1 eV [34]. After around 80 min annealing at 60 ◦C Na , Ny  Nc and Neff is almost entirely due to stable defects. If the initial dopant removal is complete or initial dopant concentration is small (with respect to to deep defects) the effective doping concentration is given by a simple relation |Neff | ≈ gc · eq . Often the irradiations follow the planned operation scenario. For example at LHC the detectors are operated at T ≈ −10◦ C for 1/3 of the year then stored for few weeks at close to room temperature and the rest of the year at T ≈ −10◦C. The corresponding temperature history of a whole year can be compressed roughly to 4 min at 80◦ C. The whole period of operation therefore consists of multiple irradiation and annealing steps, which is also referred to as CERN scenario [34]. The parameters of Hamburg model are used to predict the evolution of full depletion voltage of silicon pixel (n+ − n − p+ ) and strip detectors (p+ − n − n+ ) at LHC experiments. The agreement of predictions with measurements during LHC operation was good, as shown on few examples in Fig. 21.15.

Fig. 21.15 The agreement of predicted and measured Vf d for (a) LHCb Velo detector [42], (b) ATLAS-Insertable B layer pixel detector [43] (c) CMS—strip detectors in the outer region [44]. For (b) the prediction is denoted by black dots and measurements as bars with different colors

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As can be seen in Figs. 21.15, the agreement of Hamburg model with measurements is reasonable and allows for predictions of operation up to the end of their lifetime at LHC. It is evident that careful planning of maintenance and technical stops is required to keep Vf d as low as possible. Even though oxygen rich silicon was used for ATLAS pixel detectors, they will be operated under-depleted at least for some time at the end of LHC operation. The depleted region after space charge inversion grows from the pixel side and for Vbias < Vf d the detector performance is similar to that of somewhat thinner detector, still providing efficient tracking. On the other hand irradiated strip detectors at LHC (p+ − n − n+ ) require at all times Vbias > Vf d as the region around the strips needs to be depleted for achieving sufficient charge collection efficiency. The maximum bias voltage for e.g. ATLAS strip detectors is set to 450 V, which is sufficient for full depletion over the entire operation program before the HL-LHC upgrade. Standard float zone detectors are used for the fact that the larger fraction of damage is coming from neutrons and oxygenated detectors would therefore offer no significant advantage.

Defect Engineering The radiation tolerance of silicon can be improved by adequate defect engineering. Defect engineering involves the deliberate addition of impurities in order to reduce the radiation induced formation of electrically active defects or to manipulate the defect kinetics in such a way that less harmful defects are finally created. It has been established that enhanced concentration of oxygen in FZ detectors reduces the introduction rate of stable defects by factor of ∼3 after charged hadron irradiations (see Table 21.4). The most likely explanation is that oxygen acts like a trap for vacancies (formation of an uncharged V-O complex) and therefore prevents formation of charged multi-vacancy complexes. In addition, Oxygen is also related to formation of deep donors (see Fig. 21.6). On the opposite carbon enhances the concentration of vacancies as it traps interstitial silicon atoms and reduces the recombination. Since the concentration of oxygen is not high enough in the disordered regions-clusters, it has little or no effect after neutron irradiations. Different stable damage in neutron and charged hadron irradiated detectors at equal NIEL is an evidence of NIEL hypothesis violation. The diffusion oxygenated float zone detectors are used for the inner-most tracking detectors at LHC, where significant reduction of Vf d is required as shown in Fig. 21.15. The oxygen concentration in DOFZ detectors is around 2·1017 cm−3 , which is up to an order of magnitude lower than the oxygen concentration in Czochralski (Cz) silicon. They have only recently become available as detector grade material with resistivity (>1 k cm) high enough to allow production of 300 μm thick detectors [45]. The increase of Vf d after irradiation was found to be smaller or equal to that of DOFZ detectors as shown in Fig. 21.16a. Moreover, for n-type Cz detectors (less evident in p-type Cz) stable donors (gc ∼ −5 · 10−3 cm−1 ) are introduced instead of acceptors after fast charged hadron and γ -ray irradiations. The oxygen in form

21 Solid State Detectors for High Radiation Environments

700

standard FZ

9

O enriched FZ

8

C enriched FZ

600

MCz n-type

|Neff| [1012 cm-3]

7

500

MCz p-type

6

400

5 300

4 3

200

Vfd [V] (for W=300 μm)

10

993

2 100

1 0

0 0

2

4 6 8 Φ 24 GeV p [1014 cm-2]

10

12

350 MCz - n (300 um, 2e14 )

300

DOFZ n ( 300 um, 2e14) Epi-n (50 um, 9.7e14)

Vfd [V]

250

FZ-n (50 um,1.1e15)

200 150 100 50 0 1

10

100

1000

10000

100000

Annealing time @ 80C [min]

Fig. 21.16 (a) Influence of carbon and oxygen enrichment and wafer growth on the change of Neff as function of fluence. (b) Annealing of the Magnetic Cz-n type (MCz) and diffusion oxygenated samples after 2 · 1014 cm−2 . Also shown are thin epitaxial and standard FZ detectors irradiated to fluences around 1015 cm−2 . Note the typical behavior of detectors with positive space charge for epitaxial and MCz detectors

of a dimer [O2i ], which is more abundant in Cz than FZ detectors, is likely to be responsible. It is a precursor for formation of radiation induced shallow donors (thermal donors) [46]. The reverse annealing in Cz detectors has approximately the same amplitude as in FZ but is delayed to such extent that may not even play an important role at future experiments. The different sign of gc and gY produce a different shape of Neff annealing curve (see Fig. 21.16). During the short term annealing the Vf d increases and then starts to decrease as acceptors formed during late stages of annealing compensate the stable donors. Eventually the acceptors prevail and the Vf d starts to increase again. Another interesting material is epitaxial silicon grown on low resistivity Cz substrate [47]. Stable donors are introduced after charge hadron irradiation with

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rates depending on the thickness of the epitaxial layer (gc = −4 · 10−3 to − 2 · 10−2 cm−1 , for thickness of 150–25 μm). They exhibit also the smallest increase of |Neff | after neutron irradiations, but are only available in thicknesses up to 150 μm.

Control of Space Charge The opposite sign of gc and gY and |gY | > |gc | opens a possibility to control Vf d with a proper operation scenario and to keep it low enough to assure good charge collection (see Fig. 21.16b). This has been demonstrated with thin epitaxial detectors which were irradiated in steps to eq = 1016 cm−2 and annealed for 50 min at 80◦ C during the steps which is roughly equivalent to room temperature storage during non-operation periods at LHC or HL-LHC (see Fig. 21.17) [48]. The compensation of stable donors by acceptors activated during the irradiation steps resulted in lower Vf d after eq = 1016 cm−2 than the initial Vf d . Allowing detectors to anneal at room temperature during non-operation periods has also a beneficial effect on leakage current and trapping probability as will be shown later. The use of silicon material with opposite sign of stable damage for neutrons and charged hadrons can be beneficial in radiation fields with both neutron and charged hadron content. The stable acceptors introduced by neutron irradiation compensate stable donors from charged hadron irradiations and lead to reduction of Vf d as demonstrated in [49]. An example is shown in Fig. 21.17b for MCz n-type pad detectors which were irradiated by 23 GeV protons (open symbols) and then by neutrons (solid symbols). The additional neutron irradiation decreases the Vf d . (a )

(b)

250

Vfd [V]

200 150

50 m after 50 min@80C annealing 25 m after 50 min@80C annealing

50 m simulation

°

100 50 0 0

25 m simulation

2.1015

4.1015

6.1015 8.1015 [cm-2]

1016

eq

Fig. 21.17 (a) An example of space charge compensation through annealing in a thin epitaxial detector irradiated with 23 GeV protons to eq = 1016 cm−2 . The lines denote the Hamburg model prediction. (b) Beneficial effect of irradiations by protons and neutrons on Vf d for MCz n-type detectors

21 Solid State Detectors for High Radiation Environments

995

21.4.2.2 Electric Field The occupation probability (Eq. (21.15)) of a deep level is determined by its position in the band gap, temperature and concentration of free carriers. The occupancy of initial shallow dopants is largely unaffected by p, n, T and Neff is constant over the entire bulk. The irradiation introduces deep levels which act as generation centers. Thermally generated carriers drift in the electric field to opposite sides (bulk generation current). The concentration of holes is thus larger at the p+ contact and of electrons at the n+ contact. Some of these carriers are trapped and alter the space charge i.e. steady state Pt in Eq. (21.14). As a result the Neff is no longer uniform, but shows a spatial dependence, with more positive space charge at p+ and more negative at n+ contact Such a space charge distribution leads to an electric field profile different from linear. The electric field profile can be probed by measuring the current induced by the motion of carriers generated close to an electrode (so called Transient Current Technique). They drift over the entire thickness of detector. The measured induced current at time t after the injection, is then proportional to the electric field, at the position of the drifting charge at time t according to equation i = −q Ew · v. An example of such a measurement can be seen in Fig. 21.18b, where carriers at the back of the detector (n+ contact) are generated close to electrode by a short pulse of red light. The shape of the current depends on the voltage and temperature. At lower voltages and higher temperatures the electric field shows two peaks, which can only be explained by the space charge of different signs at both contacts. This is usually referred to as “double junction” profile [50, 51], the name indicating that the profile is such as if there were two different junctions at both contacts (p+ − n p − n+ structure). This is evident for under-depleted detectors where both junctions are separated by an un-depleted bulk. Usually one of the regions dominates spatially (b) I [arb.]

(a)

jp (x), p(x)

jn (x), n(x)

n type

p type

j(x)

+

0.5 0.4

T=-10 o C T=10 o C

280 V

+

0.3 200 V

Neff(x)

depth/thickness E(x)

1

0.2 120 V

0.1

+ −

0 0

5

10

15

20

25

t[ns]

Fig. 21.18 (a) Illustration of mechanism leading to non-uniform Neff . (b) Induced current due to drift of holes from n+ side to p+ side in 300 μm thick oxygenated detector irradiated with 23 GeV protons to 2 · 1014 cm−2

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(also called the “main junction”) which determines the predominant sign of the space charge and annealing properties. The space charge profile depends on the balance between the deep levels which occupation depends on n, p and shallow defects mostly unaffected by n, p. Apart from thermally generated carriers the non-equilibrium carriers which modify the electric field can also be generated by ionizing particles or continuous illumination of detector by light [52].

Modeling of the Field Even more precise insight in electric field, particularly for heavily irradiated detector (eq > 1015 cm−2 ), is obtained by a more elaborate technique called Edge-TCT [53] shown in Fig. 21.19, where the polished edge of the silicon strip detector is illuminated by narrow beam of infra-red light. The induced current measured promptly after light injection is proportional to the sum of the drift velocities of electrons and holes at a given depth of injection. The drift velocity profile of an detector is hence obtained by scanning over the edge of the detector at different depths. The profiles of heavily irradiated silicon detectors are shown in Fig. 21.20. The velocity profile in detector moderately irradiated with neutrons (Fig. 21.20a) deviates only slightly from simple model of constant Neff inside the bulk, while at higher fluence (Fig. 21.20b) the electric field shows typical “double junction” behavior, with some remarkable features: • the main junction penetrates deeper than expected using gc measured at low fluences • the high field region at the back extends deep into the detector • the electric field is present in the whole bulk even at very modest voltages

HV BIAS−T

T=−20 oC

z y

Miteq AM−1309

oxide polished edge

scaning direction

y=0 n+ implant

−−− − FWHM~8 μm

++ ++

drift direction electrons

drift direction holes

p+ implant y=W

aluminium NOT TO SCALE !

Fig. 21.19 The principle of the Edge-TCT technique

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(a)

2

(b) 15

eq=10

-2

cm

1.5

1

V bias=0 V V bias=100 V V bias=200 V V bias=300 V V bias=400 V V bias=500 V V bias=600 V V bias=700 V V bias=800 V

0.5

ve+vh [arb.]

ve+vh [arb.]

21 Solid State Detectors for High Radiation Environments

2

V bias=0 V V bias=100 V V bias=200 V V bias=300 V V bias=400 V V bias=500 V V bias=600 V V bias=700 V V bias=800 V

1.5

1

16 eq=1x10

cm-2

0.5

0

0 0

50

100

150

(c)

200

250 300 y (depth) [ m]

0

50

100

150

(d)

200

250 300 y (depth) [ m]

Fig. 21.20 The velocity profiles of neutron irradiated detectors to (a) eq = 1015 cm−2 , (b) eq = 1016 cm−2 and 23 GeV proton irradiates detectors to (c) eq = 1.8 · 1015 cm−2 and (d) eq = 1.7 · 1016 cm−2 . The measurements were performed with 300 μm thick ATLAS-07 prototype strip detectors with 100 μm pitch and 20 μm implant width at −20 ◦ C. Strips are at y = 0 μm

• the velocity in the neutral bulk is very high reaching almost a third of the saturation velocity at very high bias voltages The appearance of the electric field in the neutral bulk can be explained by the increase of undepleted bulk resistivity and increase of generation current. As both increase also higher field is required for transport of thermally generated carriers across the detector in a steady state. The electric field in charged hadron irradiated detectors is almost symmetrical at lower fluence (Fig. 21.20c) and becomes similar to neutron irradiated ones only at very high bias voltages (Fig. 21.20d). Already at 500 V the detector is fully active after receiving eq = 1.8 · 1015 cm−2 . The reason for such behavior is not clear, but points to higher oxygen content of the silicon wafers and different energy levels associated with changes of Neff with respect to the neutron irradiated detectors. Extraction of electric field from velocity profile is not straightforward [53], due to large uncertainties arising from saturation of drift velocity with the electric field. Instead of precisely modeling Neff (y) several key parameters can be extracted from the measured velocity profiles which can be used to constrain/anchor any electric field model, either effective or calculated from known defects. These parameters are

998

(b)

Neff 0 Neff

|E|

ve+vh [arb.]

(a)

G. Kramberger

2 vsat 1.5

1 vback

neutral bulk

0.5 vbulk y

act

0

y

act

y

0

50

back

100

150

y

back

200

250

300

y (depth) [μm]

Fig. 21.21 (a) Simplest effective space charge and electric field model in irradiated strip detectors. (b) Extraction of key parameters determining electric field from the measured velocity profile

shown in Fig. 21.21 and are: • depth of active region with negative space charge extending from the electrode side yact • velocity in undepleted bulk vbulk • depth of positive space charge region at the back of the detector W − yback • velocity at the back of the detector vback The parameters extracted for neutron irradiated detectors are shown in Fig. 21.22. The change of active region depth yact with voltage is compatible with gc up to the fluence of eq < 2 · 1015 cm−2 , while a three times lower gc was extracted at eq = 1016 cm−2 . Drift velocity in neutral bulk increases both with fluence and voltage, while the depth of the active region at the back is less dependent on fluence. It is clear that in heavily irradiated detectors (eq > 1 − 2 · 1015 cm−2 ) the Vf d doesn’t serve as a relevant parameter determining the active thickness as the whole detector becomes active with irradiation.

21.4.2.3 Charge Multiplication The increase of Neff with irradiation and high applied bias voltages lead to very high electric fields close to electrodes. They can become high enough so that the electrons gain enough energy in its free path to create new e–h pairs, a process called impact ionization. After drifting over the distance dx the number of free carriers increases by dNe,h = αe,h Ne,h dx

(21.34)

where αe,h are the impact ionization coefficients for electrons and holes [54, 55]. Charge multiplication through impact ionization is a well known process and widely

21 Solid State Detectors for High Radiation Environments

vbulk /v sat

yact [μ m]

( a) 300 250 200

0.35

15

1x10 cm-2 15 2x10 cm-2 15 5x10 cm-2 16 1x10 cm-2

0.3

0.2 0.15

100 nirr. 15 1x1015 cm-2 2x1015 cm-2 5x1016 cm-2 1x10 cm-2

50 0

0.1 0.05

100 200 300 400 500 600 700 800 900

( c)

0.8 1x1015 cm-2 2x1015 cm-2 5x1015 cm-2 16 1x10 cm-2

0.7 0.6

0

0

V bias [V]

0.5 0.4

V bias [V]

300 250 200 150

0.3

100

0.2

15

1x1015 cm-2 2x1015 cm-2 5x1016 cm-2 1x10 cm-2

50

0.1 0

100 200 300 400 500 600 700 800 900

(d) yback [μ m]

vback/v sat

(b)

0.4

0.25

150

0

999

0

100 200 300 400 500 600 700 800 900

V bias [V]

0

0

100 200 300 400 500 600 700 800 900

V bias [V]

Fig. 21.22 The relevant parameters of the electric field in the neutron irradiated silicon detector— see Fig. 21.21 for explanation

exploited in Avalanche Photo Diodes and Si-Photo-multipliers. It was however not observed directly in irradiated silicon detectors. Prediction of detector performance a decade ago based on extrapolation of damage parameters to fluences well above eq > 1015 cm−2 greatly underestimated the charge collection and detection efficiency. Part of this, better than expected, performance can be attributed to favorable electric field profile, part to smaller trapping (discussed later) and part to charge multiplication. A key factor was improved high voltage tolerance of detectors which allowed application of bias voltages exceeding 1 kV. Charge multiplication has since been undoubtedly observed with charge collection efficiency CCE > 1 in pad detectors [56], 3D detectors [57] and mostly strip detectors [58, 59] (see Fig. 21.23a). Another direct evidence came from TCT measurements where the drift of holes produced in multiplication was clearly observed as shown in Fig. 21.23b. There are several aspects of charge multiplication that make it difficult to control and master: • Charge multiplication is geometry/process dependent; fields between 15– 25 V/μm are required to produce sizable gain (∼1 e0 /μm). To achieve high gains the shape of implant and segmentation of electrodes (pitch and implant width) are very important. Strong field focusing close to implant edges leads to

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Fig. 21.23 (a) Measured charge collection dependence on voltage for 140 and 300 μm thick strip detectors. The red line denotes the charge measured in non-irradiated 140 μm thick detector. (b) Induced current pulses in strip detector for different depths of Edge-TCT injection. The second peak in the induced current pulses is due to multiplied holes drift

•

•

• •

higher gains. This is also the reasons why larger gains were observed in highly segmented detectors. Charge gain depends on the hit position within the electrode. In highly irradiated strip detectors higher gain was observed for tracks few μm away from the implant, where the electric fields are highest [60]. The holes produced in multiplication are trapped by deep defects (change of free hole concentration, p, in Eq. (21.15)) which reduce the negative space charge— act as a feedback. Therefore gain increases moderately with voltage and is usually limited to factors below 200 V using IBL readout electronics (FE-I4) [80].

21.4.2.8 Timing Detectors At HL-LHC coping with large particle fluxes emerging from collisions will be an enormous challenge. On average 200 p-p collisions will occur every 25 ns, with collision points distributed normally along the beam with σz = 5 cm and in time with σt = 180 ps. Resulting track and jet densities in the detector complicate the analysis of the underlying reactions that took place. A way to cope with that problem is separation of individual collisions also in terms of time of occurrence within each bunch crossing. This is particularly important for tracks/jets in forward direction for which the position resolution of primary vertex is much worse (∼1 mm). If tracks are not resolved in time, this can lead to false vertex merging. A timing resolution of around 30 ps with respect to the HL-LHC clock is required to successfully cope with pileup. Such an outstanding single particle timing resolution was up to recently impossible with silicon detectors. Three factors determine the timing resolution of each sensor: time walk which is a consequence of non-homogeneous charge deposition by an impinging particle, noise jitter (σj it t er = trise /(S/N)) and resolution of time-to-digital conversion. Standard silicon detectors of 300 μm are not appropriate for precise timing measurement as the integration time to collected all the charge and consequent rise time trise are large, hence the jitter. In addition fluctuations, not only of the amount of the

21 Solid State Detectors for High Radiation Environments

1011

charge (time walk correctable by e.g. constant fraction discrimination), but also of the deposition pattern (non-correctable time walk)—so called Landau fluctuations— ultimately limit the time resolution to effectively >100 ps [81]. High enough signal-to-noise S/N in thin detectors can be achieved by using so called Low Gain Avalanche Detectors (LGAD) [82]. They are based on a n++ − p+ − p − p++ structure where an appropriate doping of the multiplication layer (p+ ) leads to high enough electric fields for impact ionization (see Fig. 21.33) [82]. Gain factors in charge of few tens significantly improve the resolution of timing measurements, particularly for thin detectors. The main obstacle for their operation is the decrease of gain with irradiation, attributed to effective acceptor removal in the gain layer [41]. A comprehensive review of time measurements with LGADs is given in Ref. [83]. The most probable charge in 50 μm and 80 μm thick pad devices before and after irradiation is shown in Fig. 21.34a. As soon as multiplication layer is depleted the gain appears. At lower fluences the gain degradation at the depletion of multiplication layer (around 40 V) can be clearly seen. At higher fluences the gain appears at high bias voltages where over-depletion ensures that high enough electrical fields are reached; above eq > 1015 cm−2 the onset of multiplication is observed only at highest voltages of around 700 V. Such voltages correspond to very high average fields of 15 V/μm. At fluences eq > 2 · 1015 cm−2 the beneficial effect of multiplication layer is gone. The devices of the same design without multiplication layer show similar behavior as LGADs. The time resolution of LGADs was extensively measured in the test beams [84] and with 90 Sr electrons. It is shown in Fig. 21.34b for the 50 μm thick non-irradiated devices. At very large fluences of eq > 2 · 1015 cm−2 the gain, although lower than the initial, appears due to deep traps (see section on charge multiplication) Fig. 21.33 Schematic view of the Low Gain Avalanche Detector

X

Cathode Ring n++ p+ Avalanche Region e

h

Depletion Region

p

E

p+ Anode Ring

most probable charge [e]

1012

G. Kramberger

HPK-50-D (0e14) HPK-80-D (0e14) HPK-50-D (1e14) HPK-80-D (1e14) HPK-50-D (3e14) HPK-80-D (3e14) HPK-50-D (1e15) HPK-80-C (1e15) HPK-50-D (2e15) HPK-80-D (2e15) HPK-50-D (4e15) HPK-80-D (4e15)

100000

10000

1000 0

200

400

600

800

bias voltage [V]

Fig. 21.34 Dependence of most probable charge for irradiated LGAD devices on voltage for different thickness (50 and 80 μm). Fluences in the brackets are in [cm−2 ] [85]. Around 3000 e is expected for a 50 μm device without gain layer. (b) Time resolution and its noise jitter contribution measured for the non-irradiated 50 μm detector at different temperatures [86] (b)

HPK D50 & 50C Timing Resoluon, -20C

100 90 80 70 60 50 40 30 20 10 0

HPK 50D pre-rad HPK 50D 1e14 HPK 50D 3e14 HPK 50D 6e14 HPK 50D 1e15 HPK 50D 3e15 HPK 50D 6e15

1

10 Gain

100

Norm. Amplitude

Timing Resoluon [ps]

(a)

Comparison measured - WF2 pulse of HPK 50D 50-micron thick sensors

1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

pre-rad 6e14 neq/cm2 6e15 neq/cm2

0

0.2

0.4

0.6

0.8 1 1.2 Time [ns]

WF2: pre-rad WF2 6e14 neq/cm2 WF2 6e15 neq/cm2

1.4

1.6

1.8

2

Fig. 21.35 (a) Time resolution of irradiated LGAD detectors at different gains and fluences ([cm−2 ]). (b) Measured and simulated induced current pulse shape at different irradiation levels. Note that amplitudes were normalized to one

and the timing resolution degrades only moderately (see Fig. 21.35a). Moreover, multiplication in larger volume of the bulk results in faster rise time of the induced current which at given gain leads to better timing resolution (see Fig. 21.35b). The leakage current in LGADs follows the same equation as discussed in section on charge multiplication. Hence, the gain can be calculated from measurement of leakage current and calculated generation current [85, 86]. A lot of effort was spend in recent years to increase the radiation hardness of LGADs by mitigating the acceptor removal. The efforts concentrated to use of coimplantation of carbon [87] to multiplication layer aiming to reduce the removal constant or replacing boron with gallium, which should be more difficult to displace [87, 88].

21 Solid State Detectors for High Radiation Environments Au

Cr/Ti (forming carbides)

growth direction

grain bo undarie s

~2 mm

buffer layer (Pt,Mo)

1013

substrate side grind off

Fig. 21.36 Schematic view of the pCVD diamond detector

21.4.3 Diamond Detectors Although the specific ionization in diamond detectors is around three times smaller than in silicon, larger detector thickness, small dielectric constant, high break down voltage and negligible leakage current make them the most viable replacement for silicon in the highest radiation fields. The intrinsic concentration of carriers in diamond is extremely low (good insulator, ρ > 1016 cm). The detectors are therefore made from intrinsic diamond metallized at the back and the front (see Fig. 21.36) to form ohmic contacts. Most of the diamond detectors are made from poly-crystalline diamond grown with chemical vapor deposition technique (CVD). Recently also single crystalline (scCVD) detectors have become available. The quality of the poly-crystalline (pCVD) diamond as a particle detector depends on the grain size. The grains in this material are columnar, being smallest on the substrate side, and increasing in size approximately linearly with film thickness.3 Crystal faults at the boundaries between the grains give rise to states in the band gap acting like trapping centers. A widely used figure of merit for diamond is its charge collection distance (CCD), which is defined as CCD =

Qt . ρe−h

(21.42)

The CCD represents the average distance over which carriers drift. If CCD  W , it is equivalent to the trapping distance λe + λh .4 . After irradiation, and for pCVD detectors also before irradiation, the CCD depends on electric field, due to reduced probability for charge trapping at larger drift velocity. Only for non-irradiated scCVD detectors λe,h → ∞ and the CCD = W regardless of the bias voltage

3 For

this reason, many detectors have the substrate side etched or polished away. the trapping distance is: CCD = λe [1 −

4 The exact relation between the charge collection and λh λe W W W (1 − exp(− λe ))] + λh [1 − W (1 − exp(− λh ))].

1014

G. Kramberger

applied. Most commonly, the CCD is defined at E =1 V/μm or E =2 V/μm, although sometimes also CCD at saturated drift velocity is stated. The CCD of typically 500 μm thick diamond detectors has improved tremendously over the last 20 years. Current state of the art pCVD detectors reach up to 300 μm at 2 V/μm and are available from 6 inch wafers.

21.4.3.1 Radiation Hardness In pCVD detector the leakage current does not increase with irradiation; moreover it may even decrease, which is explained by passivation of defects at grain boundaries. The current density in high quality pCVD diamond is of order 1 pA/cm2, a value strongly dependent on the quality of metallized contacts. Irradiation decreases the CCD for both scCVD and pCVD diamonds with similar rate [89], pointing to the in-grain defects being responsible. It has been observed that exposing such an irradiated detector to ionizing radiation (1010 minimum ionizing particles/cm2) improves the charge collection efficiency of pCVD detectors by few 10%. This process is often called “pumping” or priming. The ionizing radiation fills the traps. The occupied traps become inactive, hence the effective trapping probability decreases. The traps can remain occupied for months due to large emission rates if kept in the dark at room temperatures. Once detectors are under bias the ionizing radiation leads to polarization of detectors, in the same way as in silicon, but with the polarization persisting over much longer times. The measurements of charge collection can therefore depend on previous biasing condition and relatively long times are needed to reach steady state of operation. The irradiation decreases the trapping distance of electrons and holes proportionally to the fluence. The relation can be derived by inserting the effective trapping time (Eq. (21.35)) in the expression for the trapping distance (Eq. (21.36)): 1 λe,h

=

1 λ0e,h

+ Ke,h · ,

(21.43)

where λ0 denotes the trapping distance of an unirradiated detector and Ke,h the damage constant. Assuming λe ≈ λh and λe + λh  W for simplicity reasons, CCD dependence on fluence can be calculated as: 1 1 ≈ + K . CCD CCD0

(21.44)

Although only approximate the Eq. (21.44) fits the measurements well over a large fluence range as shown in Fig. 21.37a. The extracted damage constant K (∼1/2 Ke,h ) from source and test beam data for particles of different energy and spectrum are gathered in Table 21.7. For high fluences the second term in Eq. (21.44) prevails and the scCVD and pCVD diamonds perform similarly. At  = 2 · 1016 cm−2 of 23 GeV protons the CCD ≈ 75 μm which corresponds

21 Solid State Detectors for High Radiation Environments

1015

450

pCVD (23 GeV p, strips, 1V/ μm)

16

400

scCVD (23 GeV p, strips, 1V/ μ m)

14

pCVD (reactor n, pad, 2V/μm)

350 300

pCVD (200 MeV π,pad, 1V/ μm)

250

pCVD (200 MeV π,pad, 1V/ μm)

12 10 8

200

6

150

Entries

(b) 18

Mean charge [1000 e]

CCD [μm]

( a) 500

FWHM/M.P.(pCVD,1e15 cm 600

300

4

200

2

100

0 200

0

100

150

14

) ~ 0.80

400

50 50

-2

FWHM/M.P.(scCVD) ~ 0.31

500

100

0 0

FWHM/M.P.(pCVD) ~ 1.00

700

0

10000

20000

-2

Φ [10 ] cm

30000

40000

50000

charge [e]

Fig. 21.37 (a) CCD vs. fluence of different particles. Detectors were 500 μm thick. The Eq. (21.44) is fitted to the data [89–91]. The irradiation particle, electrode geometry and electric field is given in brackets. (b) Energy loss distribution in CVD pad detectors. The value of FWHM, corrected for electronics noise, over most probable energy loss is shown Table 21.7 Charge collection distance degradation parameter for different irradiation particles [89–92] K[10−18 μm−1 cm−2 ]

70 MeV 1.76

800 MeV 1.21

23 GeV p 0.65

200 MeV π ∼3.5

Reactor neutrons ∼3 − 4

to mean charge of 2770 e0 . At lower fluences the first term dominates and for CCD0 ∼ 200 μm CCD only decreases by 15% after 1015 cm−2 of 23 GeV protons. The homogeneity of the response over the detector surface, which is one of the drawbacks of pCVD detectors, improves with fluence for pCVD as the collection distance becomes smaller than grain size. For the same reason also the distribution of energy loss in pCVD detector, initially wider than in scCVD, becomes narrower (Fig. 21.37b). The energy loss distribution in pCVD diamond is Gaussian, due to convolution of energy loss distributions (Landau) in grains of different sizes. One of the main advantages of the diamond is the fact that at close to room temperatures no annealing or reverse-annealing effects were observed. The drawbacks of grains in pCVD detectors can be largely overcome by using 3D diamond detectors [93], who share the same concept with silicon detectors (see Fig. 21.38). The vertical electrodes are produced by focused laser light which graphitizes the diamond. Whether the vertical electrode serves as cathode or anode depends on metal bias grid on the surface of the detector. Very narrow electrodes of ∼2 μm diameter can be made along 500 μm thick device with low enough resistivity to allow good contacts and doesn’t increase the noise. Such a good aspect ratio allows even smaller cell sizes than in silicon. The first tests showed >75% charge collection efficiency in 500 μm pCVD diamond detector of ganged 150 × 150 μm2 cells with bias voltages of only few tens Volts [94]. A much better homogeneity of charge collection over the surface (columns are parallel to grain boundaries) and narrower distributions of collected charge were obtained than in planar diamond detectors.

1016

G. Kramberger

Fig. 21.38 Photograph of 3D diamond strip detector in black rectangle with a square cell of 150 μm size. Strip detector of the same geometry with planar electrodes is shown below (from [93])

The main problem with diamond 3D detectors is the rate of production in particular for large area as even if laser beam is powerful enough and is split into several parallel beams. Currently the rate is limited to roughly ten thousand holes a day. The diamond detectors are used also outside particle physics for particle detection such as for fusion monitoring where neutrons are detected, for alpha particle detection, for determination of energy and temporal distribution of proton beams and in detection of ions during the teleradiology. They are also exploited for soft X-ray detection, where the solar-blindness and fast response of diamond detectors are the keys of their success.

21.4.4 Other Semiconductor Materials Silicon is in many respects far superior to any other semiconductor material in terms of collected charge, homogeneity of the response and industrial availability. Other semiconductor materials can only compete in niche applications where at least one of their properties is considerably superior or where the existing silicon detectors cannot be used. For example, if low mass is needed or active cooling can not be provided, high leakage current in heavily irradiated silicon detectors is intolerable and other semiconductor detectors must be used. The growth of compound semiconductors is prone to growth defects which are frequently unmanageable and determine the properties of detectors before and after irradiation. If a high enough resistivity can be achieved, the detector structure can be made with ohmic contacts. However, it is more often that either a Schottky contact or a rectifying junction is used to deplete the detector of free carriers. Only

21 Solid State Detectors for High Radiation Environments

1017

a few compound semiconductors have reached a development adequate for particle detectors. They are listed in the Table 21.2. For particle physics application some other very high-Z semiconductor such as CdZnTe or HgI2 are inappropriate due to large radiation length. Silicon Carbide was one of the first alternatives to silicon proposed in [95, 96]. It is grown as epitaxial layer or as bulk material. Even though at present the latter exhibits a lot of dislocations (inclusions, voids and particularly micro-pipes) in the growth and the former is limited to thicknesses around 50 μm, both growth techniques are developing rapidly and wafers are available in large diameters (10 inch). Due to the properties similar to diamond the same considerations apply as for diamond with an important advantage of 1.4 larger specific ionization (55 e-h/μm). Presently the best performing detectors are produced by using slightly n-doped epitaxial layers of ≈50 μm forming a Schottky junction.They exhibit 100% charge collection efficiency after full depletion and negligible leakage current [97]. Also detectors processed on semi-insulating bulk (resistivities ∼1011 cm) with the ohmic contacts show CCD up to 40 μm [96] at 1 V/μm for few hundred μm thick material. After irradiation with hadrons the charge collection deteriorates more than in silicon or in diamond. For epitaxially grown SiC the degradation of CCD is substantial with Ke ≈ 20 · 10−18 cm2 /μm and Kh ≈ 9 · 10−18 cm2 /μm for reactor neutron and 23 GeV p irradiated samples at high electric fields of 10 V/μm [97, 98]. The leakage current is unaffected by irradiation or it even decreases [98]. GaAs The resistivity of GaAs wafers is not high enough for the operation with ohmic contacts and detectors need to be depleted of free carriers, which is achieved by Schottky contact or a p − n junction. GaAs detectors were shown to be radiation hard for γ -rays (60 Co) up to 1 MGy [99]. As a high Z material these detectors are very suitable for detection X and γ rays. Their tolerance to hadron fluences is however limited by loss of charge collection efficiency, which is entirely due to trapping of holes and electrons. The Vf d decreases with fluence [100, 101] which is explained by removal or compensation of as grown defects by irradiation. Although larger before the irradiation, the trapping distance of electrons shows a larger decrease with fluence than the trapping distance of holes. The degradation of charge collection distance at an average field of 1 V/μm (close to saturation velocity) in 200 μm thick detectors is very large Ke,π ≈ 30 · 10−18 cm2 /μm and Kh,π ≈ 150 · 10−18 cm2 /μm [100]. One should however take into account that specific ionization in GaAs is four times larger than in diamond. The leakage current increases moderately with fluence up to few 10 nA/cm2 , much less than in silicon, and starts to saturate at fluences of around 1014 cm−2 [100, 101]. The GaAs exhibit no beneficial nor reverse annealing of any detector property at near to room temperatures.

1018

G. Kramberger

GaN The GaN detectors produced on few μm thin epitaxial layer shown charge collection degradation which is much larger than in Si [102]. Further developments in crystal growth may reveal the potential of material.

21.4.5 Comparison of Charge Collection for Different Detectors The key parameter relevant to all the semiconductor particle detectors is the measured induced charge after passage of minimum ionizing particles. The charge collection dependence on fluence in different semiconductor pad detectors is shown in Fig. 21.39a. A 3D pad detector (all columnar electrodes connected together) shows best performance, while smallest charge is induced in SiC and pCVD diamond detectors. The induced charge decreases with fluence and at most few thousand e0 can be expected at eq > 1016 cm−2 . Although pad detectors are suitable for material comparison the effect of segmentation and choice of the type of the read-out electrodes determine to a large extent the performance of the detectors. The superior charge collection performance of segmented silicon planar detectors with n+ electrodes to pad detectors can be seen in Fig. 21.39b. A signal of around 7000 e0 is induced in epitaxial p-type and Fz p-type strip detectors at eq = 1016 cm−2 . At the highest fluence shown the signal in a silicon pad detector is only half of that in a strip detector. On the other hand a device with p+ readout performs worst of all. (b)

SiC-n, 45 um, 600 V [us, RT] Epi-n, 150 um, od., [25 ns, -10C] Epi-p, 150 um, od. ,[25ns, -10C] Epi-n, 75 um, od. , [25 ns, -10C] Fz n-p, 300 um, 500 V, [25ns, -10C] MCz-p, 300 um, 500V, [us,-30C] Fz, n-p, 3D (235 um), od., IR laser aver. sc CVD diamond, 1 V/um, [us, RT] pCVD, CCD0=100 um, 500 um, 1V/um, [us,RT] FZ n-p, 300 um, 500 V spagehetti [25 ns, -10C]

30000 25000 20000 15000 10000

Fz, n-p, strips (300-80-20), 800 V, [25 ns, -30C] Fz, n-p, strip (300-80-20), 800V, [25ns,-30C] FZ, n-p, 3D pixel (230-250x50) 150V [25 ns, -30C] Fz, n-p, pixel (285-400x50), 600 V,[25 ns,-20C] Fz, n-n, pixel (300-250x250), 600V, [25 ns, -10C]

30000

Fz, n-p, pixel, (75-400x50), 600V [25 ns, -30C] MCz, n-n, strip (300-80-20), 800 V,[25ns,-30C] MCz, p-n, strip (300-80-20), 600V, [25 ns, -30C] Epi, n-p,strip (150-80-20), 800 V, [25 ns -30C] pCVD diamond, pixel (300,400x50), 500V

25000

20000 15000 10000 5000

5000 0

collected charge [e]

collected charge [e]

(a)

0

0

1

10 Φeq [1014 neq /cm2 ]

100

1000

0

1

10

100

1000

Φ eq [1014 neq/cm2]

Fig. 21.39 (a) Comparison of charge collection in different detectors and materials; given are material, thickness, voltage, [shaping time of electronics and temperature]. “od.” means at Vbias > Vf d . All detectors were irradiated with 23 GeV protons, except 75 μm epi-Si and 300 μm thick “spaghetti” diode which where irradiated with reactor neutrons. For diamond detectors the mean, not the most probable, charge is shown. (b) Charge collection in different segmented devices; the segmentation is denoted for strips (thickness-pitch-width) and pixels (thickness-cell size)]. Solid markers denote neutron irradiated and open 23 GeV proton irradiated samples

21 Solid State Detectors for High Radiation Environments

(a)

1019

(b)

16000

most probable charge [e]

2488-7

2935-3

2935-5

2935-6

2935-9

3·1015 cm-2

14000 12000 10000 1·1016 cm-2

8000 6000

26 MeV p 26 MeV p 24 GeV p 24 GeV p 24 GeV p 24 GeV p

2·1016 cm-2

4000

4·1016 cm-2

2000

8·1016 cm-2

1.6·1017 cm-2

0 0

200

400

600

800

1000

1200

1400

bias voltage [V]

Fig. 21.40 (a) Dependence of collected charge in different planar silicon detectors on voltage up to the extreme fluences at −10 ◦ C. The color bands are to guide the eye. (b) Test beam (120 GeV π) measurement of detection efficiency in heavily irradiated 3D detector with single electrode cell of 50 × 50μm2 shown in the inset [79]

The detection efficiency, however, depends on signal-to-noise ratio, which should be maximized. A choice of material, electrode geometry and thickness determine the electrode capacitance which influences the noise of the connected amplifier (Chap. 10). At given pixel/strip geometry the highest electrode capacitance has a 3D silicon detector, followed by a planar detector with n+ electrodes, due to required p-spray or p-stop isolation which increases the inter-electrode capacitance. Even smaller is the capacitance of p+ electrodes which is of 1 pF/cm order for strip detectors. The smallest capacitance is reached for diamond detectors owing to small dielectric constant.

21.4.5.1 Operation at Extreme Fluences A combination of trapping times saturation, active neutral bulk and charge multiplication allows silicon detectors to be efficient in radiation environments even harsher than that of HL-LHC, approaching those of FCC. The operation of silicon detectors was tested up to eq = 1.6 · 1017 cm−2 and is shown in Fig. 21.40a for short strip detectors with ganged electrodes (“spaghetti” diode). Detectors remained operational and most probable charge of around 1000 e was measured in 300 μm thick detectors at 1000 V. At high fluences (>2 · 1015 cm−2 ) the collected charge is linearly proportional with o bias voltage in whole range of applicable voltages and the dependence of charge on voltage and fluence can be parametrized with only two free parameters [103], Q(V , eq ) = k · V · (

eq )b , cm−2

1015

where b = −0.683 and k = 26.4 e/V for 300 μm thick detectors.

(21.45)

1020

G. Kramberger

A small cell size 3D detector (50 × 50μm2, 1E) irradiated with charged hadrons to eq = 2.5 · 1016 cm−2 was recently found to be fully efficient at voltages even below 200 V (see Fig. 21.40b) [79]. A rough simulation of collected charge in such a device based on known data predicts collected charge >3000 e0 after the fluence of 1017 cm−2 , which may be already enough also for successful tracking.

21.4.6 Radiation Damage of Monolithic Pixel Detectors The monolithic pixel detectors, which combine active element and at least first amplification stage on the same die, are widely used in x-ray and visible imaging applications. Their use as particle detectors is limited for applications where radiation environments are less severe (space applications, e+ − e− colliders), either because of small hadron fluences or because of radiation fields dominated by leptons and photons (see Fig. 21.3). The CCD is the most mature technology while CMOS active pixel sensors were successfully used for particle detection in STAR experiment at Relativistic Heavy Ion Collider over the last decade. These detectors are more susceptible to radiation damage due to their charge collection mechanism and the readout cycle. Recently several CMOS foundries offered a possibility to apply high voltage which can be used for depletion of substrate on which CMOS circuitry resides thereby enabling fast charge collection by drift. This greatly enhanced both radiation hardness of CMOS detectors and their speed. The principles of operation of these detectors were addressed in section on Solid state detectors. Here on only the aspects of radiation hardness of aforementioned detectors will be addressed.

21.4.6.1 CCDs The CCD5 is intrinsically radiation soft. The transfer of the charge through the potential wells of the parallel and serial register is very much affected by the charge loss. At each transfer the fraction of the charge is lost. The charge collection efficiency is therefore calculated as CCE = (1 − CT Ip )n × (1 − CT Is )m , where CT Is and CT Ip denote the charge collection inefficiency of each transfer in serial and parallel register. An obvious way of improving the CCE is a reduction of the number of transfers (m and n). Applications requiring high speed such as ILC, where the readout of the entire detector (n ∼ 2500) within 50 μs is neeeded, the serial register is even omitted (m = 0) and each column is read-out separately (column parallel CCD [104]). The CCDs suffer from both surface and bulk damage.

5 CCD

is often not considered to be monolithic devices.

21 Solid State Detectors for High Radiation Environments Fig. 21.41 The principle of notch CCD. An additional n+ implant creates the minimum in potential

1021

CCD

Notch CCD add. implant

oxide

buried ch. X

X

X X

X X X X X X X

X

X X X X X X X X X X

Potential profile in the channel X X X X X X X X X X X X X

X

radiation induced defects

X X X X X X X X X X X X X X

X

X

charge packet

The increase of CT I is a consequence of bulk and interface traps. There are several methods to improve the CT I : • The transport of the charge takes place several hundred nm away from the surface by using a n+ implant (buried channel), which shifts the potential minimum. The transport is less affected by trapping/detrapping process than at the interface traps. • Operation of CCDs at low temperatures leads to filling of the traps with carriers— electrons. Since emission times are long (Eq. (21.16)) the amount of active traps is reduced. • If the density of signal electrons (ns ) is larger than the trap concentration only a limited amount of electrons can be trapped, thus CT I ∝ Nt /ns . An additional n+ implant can be used for buried channel CCDs to squeeze the potential minimum to much smaller volume (see Fig. 21.41). • The CTI depends on the charge transfer timing, i.e. on the clock shapes. The transfer of the charge from one pixel to another should be as fast as possible to reduce the trapping. The choice of the clock frequency, number of the phases (2 or 3) and shape of the pulses, which all affect the CTI, is a matter of optimization (see Fig. 21.42). The transfer time from one well to another can be enhanced by an implant profile which establishes gradient of the electric field. • If traps are already filled upon arrival of the signal charge they are inactive and CTI decreases. The effect can be achieved either by deliberate injection of charges (dark charge) or by exploiting the leakage current. In the same way also the pixel occupancy affects the CTI. The radiation affects also operation of detectors due to surface effects. The surface generation current which is a consequence of interface traps is in most of the applications the dominant source of current in modern CCDs. Very rarely the bulk damage is so high that the bulk generation current dominates. The surface dark current can be greatly suppressed by inverse biasing of the Si-SiO2 interface [105]. The voltage shift due to oxide charge requires proper adjustment of the amplitude of the gate drive voltages. However the supply current and power dissipation of the gate drivers can exceed the maximum one as they both depend on the square of the voltage amplitude.

1022

G. Kramberger

Fig. 21.42 Comparison of charge transfer of 2 and 3 phase CCD (P1,P2,P3 denote gate drive voltages). Note that the potential well occupies 1/4 of the pixel volume for 2 phase CCD and up to 2/3 for 3-phase CCD. The signal charge is shaded

The CCDs can probably not sustain radiation fields larger than at the ILC (see Table 21.1), particularly because of the bulk damage caused by neutrons and high energy leptons.

21.4.6.2 Active CMOS Pixels In conventional monolithic active CMOS pixel sensors (Chap. 5) [106, 107] the n+ well collects electron hole pairs generated by an ionizing particle in the p doped epitaxial layer (see Fig. 21.43). The built-in depletion around the n+ well is formed enabling the drift of the carriers. In the major part of the detector the charge is collected from epitaxial p-type silicon through the diffusion. The charge collection process depends on epitaxial layer thickness and takes tens of ns. Above 90% of the cluster charge is induced within ∼100 ns for 15 μm thick epitaxial layer [108]. Since the n+ wells are used as collection electrodes, only nMOS transistors can be used for the signal processing circuit. The level of complexity of signal processing after the first stage depends on the CMOS technology used (number of metal layers, feature size). The charge collection by thermal diffusion is very sensitive to electrons lifetime, which decreases due to the recombination at deep levels. The loss of collection efficiency and consequently smaller signal-to-noise ratio is the key limitation for their use. The way to increase the radiation tolerance is therefore the reduction of diffusion paths. This can be achieved by using many n+ collection diodes per pixel area, which improves the charge collection efficiency. The price for that is a larger

21 Solid State Detectors for High Radiation Environments gate of the first stage FET

OXIDE S T N WELL P WELL I − + (4)

(1)

+−

Fig. 21.43 Schematic view of the radiation effects in active CMOS Pixel Detector: (1) generation of carriers—leakage current; (2) recombination of diffusing carriers; (3) positive oxide charge buildup leads to punch trough the p well; (4) charge trapping at shallow trench isolation structure

1023

+− −+ −+ +− +− +− −+ −− + + −+ −+ +− −+ −+

N+ (3)

(2)

EPITAXIAL LAYER − p type SUBSTRATE (p type)

ionizing particle track

capacitance and leakage current of the pixel. An increase of the epitaxial layer will increase the fraction of recombined charge, but the absolute collected charge will nevertheless be larger. The reduction of the collection time can be achieved by a gradual change of epitaxial layer doping concentration which establishes electric field. The generation-recombination centers give rise to the current and cooling is needed to suppress it. It increases the noise and requires more frequent reset of the pixel. The active pixel detectors were proven to achieve detection efficiencies of >95% at eq = 2 · 1012 cm−2 [109], suggesting an upper limit of radiation tolerance to hadron fluences of eq ≤ ·1013 cm−2 . The active pixel sensors are CMOS circuits and therefore susceptible to surface damage effects. Apart from the damage to transistor circuitry which is discussed in next section in some CMOS processes the n-well is isolated from p-well by shallow trench SiO2 isolation. The radiation induced interface states serve as trapping centers and reduce the signal. The active pixel sensors were shown to be tolerant to ionizing radiation doses of up to 10 kGy [110]. The damage effects discussed above are shown in Fig. 21.43.

Depleted CMOS In recent years a so called depleted CMOS or high voltage CMOS (HV-CMOS) process has become available by different foundries. These processes allow application of high voltage to the p substrate which becomes depleted. Charge collection by drift significantly improves both speed and radiation tolerance of these devices.

1024

G. Kramberger

Fig. 21.44 Schematic view of (a) small electrode and (b) large electrode HV-CMOS detectors [111]

( a)

(b) mean charge [e]

4500 nirr. 2e14 cm-2 5e14 cm-2 1e15 cm-2 2e15 cm-2 5e15 cm-2 1e16 cm-2

4000 3500

3000 2500

25 ns shaping 20 Ωcm

2000 1500 1000 500 0 0

20

40

60

80

100

120

substrate bias voltage [V]

Fig. 21.45 (a) Dependence of substrate Neff on fluence of devices produced by two different foundries on different substrate resistivities. Fit of Hamburg model to the data is shown with initial acceptor removal parameters left free [37]. (b) Charge collection in irradiated passive HV-CMOS diode array connected to LHC speed electronics [36]

The devices differ mostly in the way the collection electrode is realized. A small n+ collection electrode is beneficial (see. Fig. 21.44a) for its small capacitance. If, however, a n+ electrode is inside a large n-well the capacitance is determined by the size of n-well (see. Fig. 21.44b), but the charge collection is faster and more homogeneous. The optimum design therefore depends on the application (see Ref. [112]). Both options are under consideration for the upgrade of pixel sub-detectors at HL-LHC. Relatively high doping concentration of the substrate (from Neff =few 1012 to 15 10 cm−3 ) emphasizes the importance of effective acceptor removal with irradiation. For low resistivity substrates the removal of shallow acceptors dominates over the creation of deep ones and the effective doping concentration initially decreases with irradiation. The active/depleted thickness at given voltage increases resulting in larger collected charge. After the initial acceptors are removed the deep acceptors determine the depleted thickness regardless of the choice of initial substrate. The dependence of Neff on fluence for different initial substrate resistivities/doping is shown in Fig. 21.45a [37]. The increase of active thickness is reflected also in charge collection measurements shown in Fig. 21.45b for a low resistivity, 20 cm, device. Note that after the irradiations the contribution from the charges diffusing

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from the undepleted substrate to the depleted region vanishes and almost no charge is measured without bias. The contribution of carriers diffusing from the undepleted substrate disappears already after  ∼ 1014 cm−2 [36] which is the reason for initial drop of charge collection efficiency. Recent studies of pixelated devices established the need for metallization their backside and/or thinning them down [113]. In most processes the high voltage for depletion of the substrate is applied from the contact on top of the device (see Fig. 21.44). After irradiation the increase of resistivity of undepleted bulk can have a large impact on fraction of weighting potential traversed by the carriers and therefore induced charge. Low impedance biasing electrode (HV bias) relatively far away from the sensing electrode and long lateral drift paths of carriers in devices without back side biasing can result in smaller induced charge than expected from the active thickness.

21.5 Electronics The front-end electronics is an essential part of any detector system. The application specific integrated circuits (ASIC) are composed of analog and digital parts. The analog part usually consists of a preamplifier and a shaping amplifier, while the digital part controls the ASIC and its communication with readout chain. The fundamental building block of the circuit, transistors, can be either bipolar or fieldeffect devices. The benefits of either bipolar or field-effect transistors as the first amplifying stage are comprehensively discussed in [114] (see Chapter 6). The equivalent noise charge of the analog front end is given by ENC 2 ≈ 2 e0 (Inm + Im M 2 F ) τsh +

4 kB T (Cd + Cc )2 gm τsh

(21.46)

where τsh is shaping/integration time of the amplifier, Inm and Im the nonmultiplied and multiplied currents flowing in the control electrode of the transistor, M current multiplication factor and F excess noise factor, Cd detector capacitance, Cc capacitance of the transistor control electrode and gm the transconductance of the transistor. The transconductance measures the ratio of the change in the transistor output current vs. the change in the input voltage. The first term is also called current (parallel) noise while the second term is called voltage (series) noise [115]. The radiation of particle detectors therefore increases both parallel noise through Inm , Im and series noise through Cd . The excess noise factor is determined by the gain and effective ratio of hole and electron ionization coefficients keff [116] F = keff M + (1 − keff )(2 − 1/M)

(21.47)

1026 Fig. 21.46 Schematic view of the MOSFET leakage current (top). The standard FET design (bottom left) with parasitic current paths and enclosed transistor design (bottom right)

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drain

gate

source

+ + + + + + + +oxide ++++++++++

n+

n+ leakage current

++++ + ++

G S

G D n+ ++++ + ++

D

S n+ p+

For moderate gains of M ≈ 10 and keff < 0.01 usual for silicon tracking detectors F ∼ 2. It follows from Eq. (21.46) that a voltage noise should dominate the current noise if charge multiplication should increase the signal/noise ratio. If this in not true the current noise increases faster than the signal with bias voltage. Therefore, integration time and electrode size and design should be carefully optimized. The bipolar transistors are susceptible to both bulk and surface damage while field effect transistors suffer predominately from surface effects. It is the transconductance that is affected most by irradiation. MOSFET Advances in integrated electronics circuitry development lead to reduction of feature size to the deep sub-micron level in CMOS technology. The channel current in these transistors is modulated by the gate voltage. The accumulation of positive oxide charge due to ionizing radiation influences the transistor threshold voltage Vt h (See Fig. 21.46). For nMOS transistors the channel may therefore always be open and for pMOS always closed after high doses. This is particularly problematic for the digital part of the ASIC leading to the device failure. The operation points in analog circuits can be adjusted to some extent to accommodate the voltage shifts. The threshold voltage depends on the square of the oxide thickness and with thick oxides typical for MOS technologies in the previous decades (>100 nm) the radiation hardness was limited to few 100 Gy. At oxide thickness approaching 20 nm the relation Vt h ∝ d 2 breaks down (see Fig. 21.8b) as explained in Sect. 21.3.2.1. The deep sub-micron CMOS processes employing such thin oxides are therefore intrinsically radiation hard. The weak dependence of Vt h on dose for deep sub-micron CMOS processes is shown in Fig. 21.47a. The interface states introducing the leakage current are largely deactivated (see Sect. 21.3.2) in deep sub-micron CMOS transistors, leading to almost negligible surface current (Fig. 21.47b). Also mobility changes less than 10% up to 300 kGy. Even with transistor parameters not severely affected by radiation the use of so called enclosed transistor layout (ELT) [117] is sometimes required to eliminate the

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Fig. 21.47 (a) Threshold voltage shift of enclosed nMOS and standard pMOS transistors as a function of the total dose for a 0.25 μm technology. (b) Leakage current for the same transistors [117].

Fig. 21.48 Schematic view of Radiation-Induced Narrow Channel Effect

radiation effects on large arrays of transistors. Radiation induces transistor leakage through the formation of an inversion layer underneath the field oxide or at the edge of the active area (see Fig. 21.46). This leads to source-to-drain and inter-transistor leakage current between neighboring n+ implants. The former can be avoided by forcing all source-to-drain currents to run under the gate oxide by using a closed gate. The inter-transistor leakage is eliminated by implementing p+ guard rings. Development of dedicated libraries to implement enclosed transistors for each deep-sub micron process is often too demanding or the functionality required for a given surface doesn’t allow enclosed transistors. In such cases a so called RadiationInduced Narrow Channel Effect (RINCE) shown in Fig. 21.48 can occur. The positive charge trapped in the lateral shallow trench isolation oxide (STI) attracts electrons and opens a conductive channel through which leakage current can flow between source and drain. This current is usually small and [119] compared to the current that can flow in the main transistor and it only influences the subtreshold region of the transistor I–V curve. Even if the functionality of the chip is preserved

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Fig. 21.49 (a) Evolution of the leakage current with TID for different NMOS transistor sizes (width/length in μm), up to 1.36 MGy. The last point refers to full annealing at 100 ◦ C. The first point to the left is the pre-rad value (b) Same as (a) but showing transistor threshold voltage shift [119].

this impacts power consumption and thermal performance of the chip. At higher doses the negative charge trapped at the interface states compensates the positive space charge (NMOS transistors) and leakage current decreases (see Fig. 21.49a). Both processes of positive oxide charge and negative charge build-up at the interface states are highly dependable on dose rate, process and annealing. The increase of the transistor leakage current affected the operation of ATLAS-IBL detector [77]. Apart from parasitic leakage current the trapped oxide charges can also moderate electric field in the transistor channel particularly for narrow channel transistor where relatively larger part of the transistor is affected. If the change in threshold voltage for NMOS is small (see Fig. 21.49b), RINCE can be a bigger problem for PMOS transistors. There, positive trapped charge (holes) at the interface states adds to the positive oxide charge. As a consequence the threshold voltage and the required current to turn transistor on change significantly. As shown in Figs. 21.49 only marginal annealing effects were observed.

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Fig. 21.50 (a) Radiation-Induced Short Channel Effect—charging of spacer oxide modifies free carrier concentration in LDD (p− ) layer. (b) Annealing releases protons/hydrogen atoms/molecules to the gate oxide [120]

Radiation Induced Short Channel Effect (RISCE) appears in both PMOS and NMOS transistors with very short channel (for both ELT and open-layout transistors) and is a consequence of transistor design with so called spacer oxide shown in Fig. 21.50a. This oxide charges and affects the amount of carriers in Low Drain Doping (LDD) extension of the transistors leading to a decrease of the transistoron current during exposure. The radiation and temperature/annealing frees protons, neutral hydrogen atoms/molecules from spacer oxide that can reach the nearby gate oxide. There they depassivate Si-H bonds and by that change the threshold voltage and transistor-on current. This process is strongly dependent on (annealing) temperature. It can be avoided at low temperature operation (T < 0◦ C) and by switching off biasing at high temperatures [120]. The constant reduction of feature size in modern CMOS processes going from 0.35, 0.25, 0.13, 0.065, 0.045 μm have also other beneficial consequences. Ever shorter transistor channel lengths result in higher speed of the devices which consumes also less power particularly in the digital part. Larger transistor densities allow more complex signal processing while retaining the die size. Unfortunately at given power constraints, the basic noise parameters of bipolar and field-effect front-end transistors will not improve with the reduction of feature size [115]. Bipolar Transistor The main origin of damage in bipolar transistors is the reduction of minority carrier life time in the base, due to recombination processes at radiation-induced deep levels. The transistor amplification factor β = Ic /Ib (common emitter) decreases according to 1/β = 1/β0 + k. The pre-irradiation value is denoted by β0 and damage constant dependent on particle and energy by k. Since gm ∝ β, degradation of β leads to larger noise and smaller gain of the transistor. Thinner base regions are less susceptible to radiation damage, so faster transistors tend to be better. The choice of base dopant plays an important role. A boron doped base of a silicon transistor is not appropriate for large thermal neutron radiation fields due to the large cross-section for neutron capture (3840 barns). The kinetic energy released (2.3 MeV) to Li atoms and α particles is sufficient to cause large bulk damage [118].

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Single Event Effects (SEE) Unlike the bulk and surface damage, the single event effects are not cumulative. They are caused by the ionization produced by particles hitting certain elements in the circuit. According to the effect they have on operation they can be: • transient; Spurious signals propagating in the circuit, due to electrostatic discharge. • static; The state of memory or register bits is changed (Single Event Upset). In case of altering the bits controlling the circuit, they can disturb functionality or prevent circuits from operating (Single-event Functional Interrupt). • permanent; The can destroy the circuit permanently (Single Event Latchup). The SEE become a bigger problem with reduction of the feature size, as relatively smaller amount of ionization is required to change properties. The radiation hardening involves the use of static-RAM instead of dynamic-RAM and processing of electronics on SOI instead on silicon bulk (physical hardening). The logical hardening incorporates redundant logical elements and voting logic. With this technique, a single latch does not effect a change in bit state; rather, several identical latches are queried, and the state will only change if the majority of latches are in agreement. Thus, a single latch error will not change the bit.

21.6 Conclusions The radiation damage of crystal lattice and the surface structure of the solid state particle detectors significantly impacts their performance. The atoms knocked-off from their lattice site by the impinging radiation and vacancies remaining in the lattice interact with themselves or impurity atoms in the crystal forming defects which give rise to the energy levels in the semiconductor band-gap. The energy levels affect the operation of any detector in mainly three ways. Charge levels alter the space charge and the electric field, the levels act as generation-recombination and trapping centers leading to increase of leakage current and trapping probability for the drifting charge. The magnitude of these effects, which all affect the signal-tonoise ratio, depends on the semiconductor material used as well as on the operation conditions. Although the silicon, by far most widely used semiconductor detector material, is affected by all three, silicon detectors still exhibit charge collection properties superior to other semiconductors. Other semiconductors (e.g. SiC, GaN, GaAa, a-Si) can compete in applications requiring certain material properties (e.g. cross-section for incoming radiation, capacitance) and/or the crucial properties are less affected by the radiation (e.g. leakage current and associated power dissipation). Radiation effects in silicon detectors were thoroughly studied and allow for reliable prediction of the detector performance over the time in different irradiation fields.

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The state of the art silicon strip and pixel detectors used at experiments at LHC retain close to 100% detection efficiency for minimum ionizing particles at hadron fluences in excess of 1015 cm−2 and ionization doses of 1 MGy. The foreseen upgrade of Large Hadron Collider require hardness to even an order of magnitude larger fluences, which presently set the ultimate benchmark for operation of semiconductor particle detectors. The efforts for improving the silicon detection properties in order to meet these demanding requirements include defect engineering by adding impurity atoms, mainly oxygen, to the crystal, operation at cryogenic temperatures and placement electrodes perpendicularly to the detector surface—3D detectors. The increase of effective doping concentration with fluence together with high voltage operation lead to charge multiplication in heavily irradiated silicon detectors which in combination with electric field in the neutral bulk and saturation of effective trapping probabilities result in efficient operation in radiation environments even harsher than that at the HL-LHC. In recent years new detector technologies appeared, such as Low Gain Avalanche Detectors which offer along with position also time resolution and depleted CMOS monolithic detectors. The latter offer for the first time fully monolithic devices with fast response and sufficient radiation hardness. The initial dopant removal plays a crucial role in performance of both LGADs and depleted CMOS detectors. Among other semiconductors diamond is the most viable substitute for silicon in harsh radiation fields, particularly with the advent of 3D diamond detectors. The silicon detector employed in less severe environments e.g. monolithic active pixels, charge coupled devices, silicon drift detectors are optimized for the required position and/or energy resolution and the radiation effects can be well pronounced and even become the limiting factor already at much lower doses. Longer drift and/or charge integration times increase the significance of leakage current, charge trapping and carrier recombination. The silicon-silicon oxide border and the oxide covering the surface of silicon detectors and electronics is susceptible to ionizing radiation. The positive charge accumulates in the oxide and the concentration of interface states, acting as trapping and generation-recombination centers, increases. These effects can be effectively reduced in silicon detectors by proper processing techniques. Thin oxides (80 >30 20.5 129

ILC-500 500 1.79 × 1034 5 727 1312 554 474 5.9 300 >80 >30 33.4 164

CLIC 3000 5.9 × 1034 50 0.24 312 0.5 40 1 44 >80 >30 ≈50 589

FCC-ee 240 8.5 × 1034

328/beam 994 13,748 36 3150 0 0 97.756 308

the last year significant progress was made on demonstrating the CLIC technology (see [8] and references therein). However a major limitation remains the lack of a significant demonstration setup, which would allow full system tests in a sizeable installation. In recent years efforts to study a circular collider option have intensified. Both at CERN and in China designs are being developed for a circular collider, based in a tunnel of about 100km in circumference, which could host an electron positron collider. The technology for such a collider is available and does not provide unsurmountable challenges, apart from the scale of the project. A design study is currently ongoing, led by CERN, to develop a conceptual design report for a such a collider hosted in the Geneva area [9]. A similar study, CEPC, is led by IHEP in Beijing, about hosting this collider in China [11]. A circular collider at the Higgs threshold would be able to deliver integrated luminosities which are— at the Higgs production threshold—higher by a factor >5 for the same running time and one interaction region, as a linear collider. It could also serve more than one interaction region simultaneously in the recirculating beams, adding up the integrated luminosities of each installed experiment. This is a big advantage over a linear collider, where the colliding beams are used only once and disposed off in beam dumps after the collision. On the other hand, the infrastructure for a 100 km installation becomes very challenging, and the energy reach of a circular machine is limited due to the losses by synchrotron radiation. It is clear that any electronpositron collider that goes beyond 350 GeV has to be linear. In that respect, linear colliders do scale with energy while circular colliders do not. For the experimenter however the challenges at any of the proposed electronpositron collider facilities are similar. The biggest difference between the proposals is the distance between bunches. At the ILC and FCC-ee (at the Higgs threshold) this time difference is with a few 100 ns very benign. At CLIC bunch distances at sub-ns

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level are anticipated, and pose additional challenges to the experiment. Nevertheless, the goals for all facilities are the same: the experiment should be able to do precision physics, even for hadronic final states, should allow the precise reconstruction of charged and neutral particles, of secondary vertices. It has to function with the very large luminosity proposed for these machines, including significant backgrounds from beam-beam interactions.

22.4.1 Physics at an LC in a Nutshell The design of a detector at a large facility like the ILC or CLIC can not be described nor understood without some comprehension about the type of measurements which will be done at this facility. A comprehensive review of the proposed physics program at the ILC facility can be found in [23, 25, 26], a review of CLIC physics is available under [7, 8]. The discussion in this section concentrates on the physics which can be done at a facility with an energy below 1 TeV. In recent years, the physics reach of a facility operating at around 250 GeV has been closely scrutinised, both at ILC and at CLIC (which is proposed to run in an initial energy stage at 380 GeV). Earlier studies have looked at the science case for a 500 GeV machine, and have explored the additional measurements possible if an energy upgrade up to 1 TeV might be possible. At a center-of-mass energy of 250 GeV the ILC will be able to create Higgs bosons in large numbers, mostly in the so called Higgs-Strahlungs process. Here a Higgs boson is produced associated to a Z boson. The great power of this process is that by reconstructing the Z, and knowing the initial beam energies, one can reconstruct the properties of the Higgs boson without ever looking at the Higgs boson itself. Thus a model independent and decay mode-blind study of the Higgs particle will become possible. In addition through the reconstruction of exclusive final states for the Higgs particle, high precision measurements of the branching ratios will be possible. On its own, precisions on the most relevant branching ratios of around 1% will be possible. Combined with the results from the LHC, this precision can be pushed to well below the percent level. Samples of the heavy electroweak bosons, W and Z, a focus of the program at the LEP collider, will in addition be present in large numbers, and might still present some surprises if studied in detail. If the energy of the facility can be increased to above 350 GeV, top-quark pairs can be produced thus turning the ILC into a top factory. Again, due to the cleanliness of the initial and the final states, high precision reconstruction of the top and its parameters will become possible. A precision scan of the top pair production threshold would determine the top mass with a statistical error of 27 MeV [27], which relates to a relative precision of ≈0.015%, far better than what can be done at LHC. Operating at 500 GeV or slightly above, the ILC will gain access to the measurement of the top-Higgs coupling, and start to become sensitive to a measurement

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of the Higgs self coupling. This latter experiment might provide evidence for the existence of this interaction at 500 GeV, but would vastly profit from even higher energies. At 1 TeV the Higgs self coupling could be measured to within 10%, which allows for reconstructing the Higgs potential and, thus, testing a cornerstone of the predictions of the standard model and the Higgs sector. Together these measurements will allow a complete test of the Higgs sector, and thus a in-depth probe of the standard model in this unexplored region. There are good reasons to assume that the standard model is only an effective low-energy theory of a more complex and rich theory. A very popular extension of the standard model is supersymmetry, which predicts many new states of matter. Even though the LHC sofar has not found any evidence for supersymmetry, many models exists which predict new physics in a regime mostly invisible to the LHC. Together ILC and LHC would explore essentially the complete phase space in the kinematic regime accessible at the energy of the ILC. Should a new state of matter be found at either the LHC or the ILC, electronpositron collisions would allow to study this sign of new physics with great precision. In addition to direct signs of new physics, as represented by new particles, the ILC would allow to indirectly explore the physics at the Terascale through precision measurements, up to energy scales which in many cases are equivalent if not higher than those at the LHC. It might well be, if no new physics is found at the LHC, that these precision measurements at comparatively low energies are our only way to learn more about the high-energy behaviour of the standard model, and to point at the right energy regime where new physics will manifest itself. Even though the ILC has been at the focus of the discussions in this chapter, all other electron-positron collider options will have a very similar physics reach—for those energies which are reachable at each facility.

22.5 Experiments at a Lepton Collider As discussed in the previous section, high energy lepton collisions offer access to a broad range of scientific questions. A hallmark of this type of colliding beam experiments is the high precision accessible for many measurements. A detector at such a facility therefore has to be a multi-purpose detector, which is capable to look at many different final states, at many different signatures and topologies. In this respect the requirements are similar to the ones for a detector at a hadron collider. The direction in which an lepton collider detector is optimised however is very different. Lepton collider detectors are precision detectors—something which is possible because the lepton collider events are comparatively clean, backgrounds are low, and rates are small compared to the LHC. The collision energy at the lepton collider is precisely known for every event, making it possible to measure missing mass signatures with excellent precision. This will make it possible to measure masses of supersymmetric particles with precision, or, in fact, masses of

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any new particle within reach of the collider. The final states are clean and nearly background-free, making it possible to determining absolute branching ratios of essentially every state visible at the lepton collider. The reconstruction also of hadronic final states is possible with high precision, opening a whole range of states and decay modes which are invisible at a hadron machine due to overwhelming backgrounds. This results in a unique list of requirements, and in particular on very high demands on the interplay between different detector components. Only the optimal combination of different parts of the detector can eventually deliver the required performance. Many of the interesting physics processes at an LC appear in multi-jet final states, often accompanied by charged leptons or missing energy. The reconstruction of the invariant mass of two or more jets will provide an essential tool for identifying and distinguishing W ’s, Z’s, H ’s, and top, and discovering new states or decay modes. To quantify these requirements the di-jet mass is ofter used. Many decay chains of new states pass through W or Z bosons, which then decay predominantly into two jets. To be able to fully reconstruct these decay chains, the di-jet mass resolution should be comparable or better than the natural decay width of the parent particles, that is, around 2 GeV for the W or Z: Edi−j et α σm = √ , = Edi−j et M E(GeV)

(22.1)

where E denotes the energy of the di-jet system. With typical di-jet energies of 200 GeV at a collision energy of 500 GeV, α = 0.3 is a typical goal. Compared to the best existing detectors this implies an improved performance of around a factor of two. It appears possible to reach such a resolution by optimally combining the information from a high resolution, high efficiency tracking system with the ones from an excellent calorimeter. This so called particle flow ansatz [28, 29] is driving a large part of the requirements of the LC detectors. Table 22.3 summarises several selected benchmark physics processes and fundamental measurements that make particular demands on one subsystem or another, and set the requirements for detector performance.

22.5.1 Particle Flow as a Way to Reconstruct Events at a Lepton Collider Particle flow is the name for a procedure to optimally combine information from the tracking system and the calorimeter system of a detector, i.e. to fully reconstruct events. Particle flow has been one of the driving forces in the optimisation of the detectors at a Lepton Collider. Typical events at the LC are hadronic final states with Z and W particles in the decay chain. In the resulting hadronic jets, typically around 60% of all stable

Physics process ZH H H Z → q qb ¯ b¯ ZH → ZW W ∗ ν ν¯ W + W − ZH → 1+ 1− X μ+ μ− (γ ) ZH + H νν → μ+ μ− X ¯ cc, H Z, H → bb, ¯ gg bb¯

Measured quantity Triple Higgs coupling Higgs mass B(H → W W ∗ ) σ (e+ e− → ν ν¯ W + W − ) Higgs recoil mass luminosity weighted Ecm B(H → μ+ μ− ) Higgs branching fractions b quark charge asymmetry Vertex detector

Tracker

Critical system Tracker and calorimeter

Table 22.3 Sub-Detector performance needed for key LC physics measurements (from [30])

Charged particle momentum resolution, pt /pt2 Impact parameter, δb

Critical detector characteristic Jet energy resolution, E/E

5 μm ⊕ 10 μm/p(GeV/c) sin3/2 θ

5 × 10−5

√ 3 ∼ 4 % or 30%/ E

Required performance

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particles are charged, slightly less than 30% are photons, only around 10% are neutral long lived hadrons, and less than 2% are neutrinos. At these energies charged particles are best re-constructed in the tracking system. Momentum resolutions which are reached in detectors are δp/p2 ≈ 5 × 10−5 GeV−1 , much better than any calorimeter system √ at these energies. Electromagnetic energy resolutions are around δEem /E = 0.15/ E(GeV), typical resolutions achieved with a good hadronic √ calorimeter are around δEhad /E = 0.45/ E(GeV). Combining these with the proper relative weights, the ultimate energy resolution achievable by this algorithm is given by σ 2 (Ej et ) = wt r σt2r + wγ σγ2 + wh0 σh20 ,

(22.2)

where wi are the relative weights of charged particles, photons, and neutral hadrons, and σi the corresponding resolution. Using √the above mentioned numbers an optimal jet mass resolution of δE/E = 0.16/ E(GeV ) can be reached. This error is dominated by the contribution from the energy resolution of neutral hadrons, √ assumed to be 0.45/ E(GeV ). This formula assumes that all different types of particles in the event can be individually measured in the detector. This implies that excellent spatial resolution in addition to the energy resolution is needed. Thus fine-grained sampling calorimeters are the only option currently available which can deliver both spatial and energy resolution at the same time. This assumption is reflected in the resolution numbers used above, which are quoted for modern sampling type calorimeters. Even though an absorption-type calorimeter— for example a crystal calorimeter as used in the CMS experiment—can deliver better energy resolution, it falls significantly behind in the spatial resolution, thus introducing a large confusion term in the above equation. Formula 22.2 describes a perfect detector, with perfect efficiency, no acceptance holes, and perfect reconstruction in particular of neutral and charged particles in the calorimeter. In reality a number of effects result in a significant deterioration of the achievable resolution. If effects like a final acceptance of the detector, missing √ energy e.g. from neutrinos etc. is included, this number easily increases to 25%/ E [31]. All this assumes that no errors are made in the assignment of energy to photons and neutral hadrons. The optimisation of the detector and the calorimeter in particular has to be done in a way that these wrong associations are minimised. From the discussion above it is clear that three effects are of extreme importance for a detector based on particle flow: as good hadronic energy resolution as possible, excellent separation of close-by neutral and charged particles, and excellent hermeticity. It should also be clear that the ability to separate close-by showers is more important than ultimate energy resolution: it is for this reason that total absorption calorimeters, as used e.g. in the CMS experiment, are not well suited for the particle flow approach, as they do not lend themselves to high segmentation. Existing particle flow algorithms start with the reconstruction of charged tracks in the tracking system. Found tracks are extrapolated into the calorimeter, and linked with energy deposits in there. If possible, a unique assignment is made between a track and an energy deposit in the calorimeter. Hits in the calorimeter belonging

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to this energy deposit are identified, and are removed from further considerations. The only place where the calorimeter information is used in the charged particle identification is in determining the type of particle: calorimeter information can help to distinguish electrons and muons from hadrons. A major problem for particle flow algorithms are unassigned clusters, and mis-assignments between neutral and charged deposits in the calorimeter. The currently most advanced particle flow algorithm, PandoraPFA, tries to minimise these effects by a complex iterative procedure, which optimises the assignments, goes through several clean-up steps, and tries to also take the shower sub-structure into account [31]. What is left in the calorimeter after this procedure is assumed to have come from neutral particles. Clusters in the calorimeter are searched for and reconstructed. With a sufficiently high segmentation both transversely and longitudinally, the calorimeter will be able to separate photons from neutral hadrons by analysing the shower shape in three dimensions. A significant part of the reconstruction will be then the reconstruction of the neutral hadrons, which leave rather broad and poorly defined clusters in the hadronic calorimeter system. Particle flow relies on a few assumptions about the event reconstruction. For it to work it is important that the event is reconstructed on the basis of individual particles. It is very important that all charged tracks are found in the tracker, and that the merging between energy deposits in the calorimeter and tracks in the tracker is working as efficiently as possible. Errors in this will quickly produce errors for the total energy, and in particular for the fluctuations of the total energy measured. Not assigning all hits in the calorimeter to a track will also result in the creating of additional neutral clusters, the so called double counting of energy. Reconstructing all particles implies that the number of cracks and the holes in the acceptance should be minimised. This is of particular importance in the very forward direction, where the reconstruction of event properties is complicated by the presence of backgrounds. However, small errors in this region will quickly introduce large errors in the total energy of the event, since many processes are highly peaked in the forward direction. In Fig. 22.5 the performance of one particular particle flow algorithm, PandoraPFA [31] is shown, as a function of the dip angle of the jet direction, cos θ . The performance for low energies of the jets, 45 GeV is close to the optimally possible resolution if the finite acceptance of the detector is taken into account. At higher energies particles start to overlap, and the reconstruction starts to pick up errors in the assignment between tracks and clusters. This effect, called confusion, will deteriorate the resolution, and will increase at higher energies. Jets at higher energies are boosted more strongly, resulting in smaller average distances between particles in the jet. This results in a worse separation of particle inside the jet, and thus a worse resolution. Figure 22.6 shows an event display of a simulated hadronic jet in the ILD detector concept for the ILC with particle flow objects reconstructed by PandoraPFA. The benefit of a highly granular detector system is clearly visible. Over the last 10 years, the Pandora algorithm has matured into a robust and stable algorithm. It is now used not only in the linear collider community, but also in long baseline neutrino experiments, and is under study at the LHC experiments.

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Fig. 22.5 The jet energy resolution, α, as a function of the dip angle | cos θq | for jets of energies from 45 GeV to 250 GeV

Fig. 22.6 Simulated jet in the ILD detector, with particle flow objects reconstructed by the Pandora algorithm shown in different colors

22.5.2 A Detector Concept for a Lepton Collider Over the years a number of concepts for integrated detectors have been developed for use at a lepton collider [32–36]. Broadly speaking two different models exist: one based on the assumption, that particle flow is the optimal reconstruction technique, the other not based on this assumption. Common to all proposals is that both the tracking system and the calorimeter systems are placed insides a large superconducting coil which produces a large magnetic field, of typically 3–5 T. Both concepts use high precision tracking and vertexing systems, inside solenoidal

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fields, which are based on state of the art technologies, and which really push the precision in the reconstruction of the track momenta and secondary vertices. Differences exist in detail in the choice of technology for the tracking devices, some rely heavily on silicon sensors, like the LHC detectors, others propose a mixture of silicon and gaseous tracking. The calorimeters are where these detectors are most different from current large detectors. The detectors based on the particle flow paradigm propose calorimeters which are more like very large tracking systems, with material intentionally introduced between the different layers. Systems of very high granularity are proposed, which promise to deliver unprecedented pictures of showering particles. Another approach is based on a more traditional pure calorimetric approach, but on a novel technology which promises to eventually allow the operation of an effectively compensated calorimeter [34]. At the ILC, detectors optimised for particle flow have been chosen as the baseline. The two proposed detector concepts ILD [32] and SiD [33] differ in the choice of technology for the tracking detectors, and on the overall emphasis based on particle flow performance at higher energies. However, both detectors have been optimised for collision energies of less than 1 TeV, while within the CLIC study the detector concepts have been further evolved to be optimised for operation at energies up to 3 TeV [35]. A conceptual picture of the ILD detector, as proposed for the ILC, is shown in Fig. 22.7. Visible are the inner tracking system, the calorimeter system inside the coil, the large coil itself, and the iron return yoke instrumented to serve as a muon identification system. A cut view of a quadrant with the sub-systems of ILD is shown in Fig. 22.8.

Fig. 22.7 Three-dimensional view of a proposed detector concept for the ILC, the ILD detector [32] (credit Ray Hori, KEK)

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Fig. 22.8 Cut through the ILD detector in the beam plane, showing one quarter of the detector [37]

22.6 Detector Subsystems A collider detector has a number of distinct sub-systems, which serve specific needs. In the following the main systems are reviewed, with brief descriptions of both the technological possibilities, and the performance of the system.

22.6.1 Trends in Detector Developments Detector technologies are rapidly evolving, partially driven by industrial trends, partially itself driving technological developments. New technologies come into use, and disappear again, or become accepted and well-used tools in the community. A challenge for the whole community is that technological trends change faster than ever, while the design, construction and operation cycles of experiments become longer. Choosing a technology for a detector therefore implies not only using the very best available technology, but also one which promises to live on during the expected lifetime and operational period of the experiment. An example of this are Silicon technologies, which are very much driven by the demands of the modern consumer electronics industry. By the times Si detectors are operational inside an experiment, the technology used to built them is often already outdated, and replacements or extensions in the same technology are difficult to get. Even more so than to the sensors this applies to readout and data acquisition systems.

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Because of the rapid progress in semiconductor technology, feature sizes in all kind of detector are getting ever smaller. Highly integrated circuits allow the integration of a great deal of functionality into small pixels, allowing the pixellation of previously unthinkable volumes. This has several consequences: the information about an event, a particle, a track, becomes ever larger, with more and more details at least potentially available and recorded. More and more the detection of particles, of properties of particles, rely no longer on averaging its behaviour over a volume large compared to the typical distances involved in the process used to measure the particle, but allows the experimenter to directly observe the processes which eventually lead to a signal in the detector. Examples of this are e.g. the Si-TPC (Silicon readout Time Projection Chamber, described in more detail below) where details of the ionisation process of a charged particle traversing a gas volume can be observed, or the calorimeter readout with Si-based pixellated detectors, given unprecedented insights into the development of particle showers. Once the volume read out becomes small compared to the typical distances involved in the process which is being observed, a digital readout of the information can be contemplated. Here, only the density of pixels is recorded, that is, per pixel only the information whether or not a hit has occurred, is saved. This results in potentially a much simpler readout electronics, and in more stable and simpler systems. These digital approaches are being pursued by detectors as different as a TPC and a calorimeter. Increasing readout speed is another major direction of developments. It is coupled but not identical to the previously discussed issue of smaller and smaller feature sizes of detectors. Because of the large number of channels, faster readout systems need to be developed. An even more stringent demand however comes from the accelerators proposed, and the luminosities needed to make the intended experiments. They can only be used if data are readout very quickly, and stored for future use. To give a specific example: the detector with the largest numbers of pixels ever built so far (until the Phase-II upgrades of the LHC detectors) has been the SLD detector at SLAC which operated during the 1990. The vertex detector, realised from charged coupled sensors with some 400 Million channels, was readout with a rate of around 1 MHz. For the ILC readout speeds of at least 50 MHz, maybe even more, are considered, to cope with larger data rates and smaller inter-bunch spacings. Technological advances in recent years have made it feasible to consider the possibility to do precision timing measurements with semi-conductor detectors. Timing resolutions in the range of 100 ps or better are becoming feasible, something completely unthinkable only a few years ago. This capability—somewhat orthogonal to the readout speed discussed above—can significantly extend the capabilities of semiconductor tracker, into the direction of so-called 4D tracking or calorimeter systems. Timing information at this level of precision can be used to measure the mass of particles through time-of-flight, and can help to separate out-of-time background from collision related events. For many applications, particularly at the LHC, radiation hardness is at a premium. Major progress has been made in recent years in understanding damage mechanisms, an understanding, which can help to design better and more radiation hard detectors. For extreme conditions novel materials are under investigation.

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22.6.2 Vertex Detectors: Advanced Pixel Detectors Many signals for interesting physics events include a long lived particle, like e.g. a B or charmed hadron, with typical flight distances in the detector from a few 10 μm to a few mm. The reconstruction of the decay vertices of these particles is important to identify them and to distinguish their decay products from particles coming from the primary vertex, or to reconstruct other event quantities like vertex charge. To optimally perform these functions the vertex detector has to provide high precision space points as close as possible to the interaction point, has to provide enough space points, so that an efficient vertex reconstruction is possible over the most relevant range of decay distances, of up to a few cm in radius, and present a minimal amount of material to the particle so as to not disturb their flight path. Ideally, the vertex detector also offers enough information that stand-alone tracking is possible based only on vertex detector hits. At the same time a vertex detector has to operate stably in the beam environment. At a hadron collider it has to stand huge background rates, and cope with multiple interactions. At a lepton collider, very close to the interaction point a significant number of beam background particles may traverse the detector, mostly originating from the beam–beam interaction. These background particles are bent forward by the magnetic field in the detector. The energy carried away by this beamstrahlung may be several 10 TeV, which, if absorbed by the detector, would immediately destroy the device. The exact design of the vertex detector therefore has to take into account these potential backgrounds. At a hadron collider, the largest challenge will be to design the detector such that it can survive the radiation dose and is fast enough and has small enough pixels to cope with the large particle multiplicity. Here pixel size, readout speed, and radius of the detector are the main parameters which need to be optimised. At a lepton collider, both size and magnetic field can be used to make sure that the detector stays clear of the majority of the background particles. The occupancy at any conceivable luminosity is not driven by the physics rate, but only by the background events. Since they are much softer than the physics events, a strong magnetic field can be used to reduce the background rates, and allow small inner radii of the system. Nevertheless, the remaining hits from beam background particles dominate the occupancies, especially at the innermost layers of a vertex detector, and therefore require fast read-out speeds. The particular time structure of the collider has an important impact on the design and the choice of the technology. At the ILC collisions will happen about every 300 ns to 500 ns, in a train of about 1 ms length, followed by a pause of around 200 ms. About 1300 bunches are expected to be in one train. A fast readout of the vertex detector is essential to ensure that only a small number of bunches are superimposed within the occupancy of the vertex detector. At CLIC the inter-bunch spacing is much smaller, putting a premium on readout speed. At LHC the typical time between collisions in a bunch crossing is order 100 ps decreasing to about 10 ps at the high luminosity LHC-HL.

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A Si-pixel based technology is considered the only currently available technology which can meet all these requirements. A small pixel size (< 20 × 20 μm2 ) combined with a fast read out will ensure that the occupancy due to backgrounds and from expected signals together remain small enough to not present serious reconstruction problems. It also allows for a high space point resolution, and a true three dimensional reconstruction of tracks and vertices essentially without ambiguities. Several silicon technologies are available to meet the demands. Increasingly, sensors based on the CMOS process are considered. Most recently devices with intrinsic gain larger than one are studied intensely, as they promise excellent performance combined with very good timing properties. Quite a number of different technologies are currently under study. Broadly they can be grouped into at least two categories: those which try to read the information content as quickly as possible, and those which try to store information on the chip, and which are readout during the much longer inter-bunch time window. Another option under study is a detector with very small pixels, increasing the number of pixels to a point where even after integration over one full bunch train the overall occupancy is still small enough to allow efficient tracking and vertexing. A fairly mature technology is the CCD technology [38, 39], which for the first time was very successfully used at the SLD experiment at the SLC collider at SLAC, Stanford. Over the past decade a number of systems based on this concept have been developed. Newer approaches use the industrial CMOS process to develop monolithic active pixel sensors (MAPS) that are at the same time thin, fast, and radiation hard enough for particle physics experiments [40]. A smaller scale application of this technology is a series of test-beam telescopes, based on the Mimosa families of chips [41], built under the EUDET and AIDA European programs [42, 43] and operated a CERN, DESY and SLAC. The Phase-II upgrade of the ALICE experiment at the LHC contains a new inner tracking system that will be completely based on the CMOS-MAP sensor ALPIDE [44]. With a pixel size of 24.9 μm ×29.3 μm, a spatial resolution of ≈ 5 μm and a time resolution of 5–10 μs is envisaged for hit rates of about 106 /cm2 /s. The CBM experiment, planned for the FAIR heavy-ion facility in Darmstadt, foresees to use MAPS for the microvertex detector. It will be based on the MIMOSIS chip, that is an advancement of the ALPIDE chip with similar pixel size and spatial resolution, but that has to cope with a much higher event rate of about 108/cm2 /s (and the associated radiation load) at the cost of a higher power consumption. The MIMOSIS chip already aims for a higher readout speed of about 5 μs. In Fig. 22.9 a measured point resolution achieved with the CMOS-MAPS technology in a test beam experiment is shown [49]. Other technologies are at a similar level of testing and verifying individual sensors for basic performance. Studies are underway to push the CMOS-MAPS towards even higher readout speeds [45]. The two parameters that currently govern the process are the time required for the pixel address encoding and the signal shaping during the preamplification. Changing the algorithm of the pixel address encoding and increasing the internal clock, could lead to an improvement from 50 ns to 25 ns for this step.

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The signal shaping currently takes about 2 μs and could be shortened to about 500 ns at the price of increasing the pixel current and therefore also increasing the power consumption. However, as the detectors at a linear collider would be operated in power-pulsing mode, the impact on the cooling requirements would be minor. Such an optimised CMOS detector for the ILC would have a readout speed of about 1 μs, i.e. it could be read out every two to three bunch crossings. Other groups explore the possibility to store charge locally on the pixel, by including storage capacitors on the pixel. Up to 20 timestamped charges are foreseen to be stored, which will then be readout in between bunch trains. The most recent example of a pixel detector at a lepton collider has been the pixel detector for the Belle-II experiment. This system is based on the DEPFET technology [46]. Charge generated by the passage of a charged particle through the fully depleted sensitive layer is collected on the gate of a DEPFET transistor, implemented into each pixel. DEPFET sensors can be thinned to remove all silicon not needed for charge collection, to something like 50 μm, or 0.1% of a radiation length. This makes this technology well suited for lepton collider applications, where minimal material is of paramount importance [48]. A problem common to all technologies considered is the amount of material present in the detector. A large international R&D program is under way to reduce significantly the material needed to build a self-supporting detector. The goal, driven by numerous physics studies, and the desire for ultimate vertex reconstruction, is a single layer of the detector which in total presents 0.1% of a radiation length,

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including sensor, readout and support. This can only be achieved by making the sensors thin, and by building state-of-the-art thin and light weight support structures. To compare, at the LHC the total amount of material present in the silicon based trackers is close to 2 radiation length, implying that per layers, close to 10% of a radiation length is present. Very thin sensor layers are possible with technologies based on fully depleted sensors. Since here only a thin layer of the silicon is actually needed for the charge collection the rest of the wafer can be removed, and the sensor can be thinned from typically 300 μm, used e.g. in the LHC experiments, to something like 50 μm or less. Several options are under study how such thin Si-ladders can then be supported. One designs foresees that the ladders be stretched between the two endcaps of the detectors, being essentially in the active area without additional support. Another approach is to study the use of foam material to build up a mechanically stiff support structure. Carbon foam is a prime candidate for such a design, and first prototype ladders have come close to the goal of a few 0.1% X0 [47]. Another group is investigating whether Si itself could be used to provide the needed stability to the ladder. By a sophisticated etching procedure stiffening ribs are built into the detector, in the process of removing the material from the backside, which will then stabilise the assembly. This approach has been successfully implemented for the vertex detector at the Belle-II experiment [48]. Material reduction is an area where close connections exist between developments done for the ILC and developments done for the LHC and its upgrade. In both cases minimum material is desired, and technologies developed in the last few years for the ultra-low material ILC detector are of large interest to possible upgrade detectors for LHC and LHC Phase-II. The readout of these large pixel detectors present in itself a significant challenge. On-chip zero-suppression is essential, but also well established. Low power is another important requirement, consistent with the low mass requirement discussed above. Only a low power detector can be operated without liquid cooling, low mass can only be achieved without liquid cooling. It has been estimated that the complete vertex detector of an ILC detector should not consume on average more than 100 W, if it is to be cooled only through a gas cooling system. Currently this is only achievable if the readout electronics located on the detector is switched off for a good part of the time, possible with the planned bunch structure of the ILC. However such a large system with pulsed power has never been built, and will require significant development work. It should not be forgotten that the system needs to be able to operate in a large magnetic field, of typically 4 T. Each switching process therefore, which is connected with large current flows in the system, will result in large Lorentz forces on the current leads and the detectors, which will significantly complicate the mechanical design of the system. Nevertheless, with current technologies power pulsing is the only realistic option to achieve the desired low power operation, and thus a central requirement for the low mass design of the detector. In Fig. 22.10 the conceptual layout of a high precision vertex detector is shown.

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Fig. 22.10 Top: Concept of a double-layer vertex detector system developed within the PLUME project. Bottom: Vertex detector for the ILD concept, based on a layout with three double layers [37]

One of the key performance figures of a vertex detector is its capability to tag heavy flavours. At the ILC b-quarks are an important signature in many final states, but more challenging are charm quarks as the are e.g. expected in decays of the Higgs boson. Obtaining a clean sample of charm hadrons in the presence of background from bottom and light flavour is particularly difficult. Already at the SLC and the LEP collider, the ZVTOP [50] algorithm has been developed and used successfully. It is based on a topological approach to find displaced vertices. Most tracks originating from heavy flavour decays have relatively low momenta, so excellent impact parameter resolution down to small (≈1GeV) energies is essential. On the other hand, due to the large initial boost of the heavy hadrons, the vertices can be displaced by large distances, up to a few cm away from the primary vertex, indicating that the detector must be able to reconstruct decay vertices also at large distances from the interaction point. The algorithms have been further developed and adapted to the expected conditions at a linear collider [51]. The performance of a typical implementation of such a topological vertex finder is shown in Fig. 22.11.

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Fig. 22.11 Purity versus efficiency curve for tagging b-quarks (red points) and c-quarks (green points) and c-quarks with only b-quark background (blue points) obtained in a simulation study for Z-decays into two (left) and six (right) jets, as simulated in the ILD detector [37]

22.6.3 Solid State Tracking Detectors: Strip Detectors To determine the momentum of a charged particle with sufficient accuracy, a large volume of the detector needs to be instrumented with high precision tracking devices, so that tracks can be reliably found and their curvature in the magnetic field can be well measured. Cost and complexity considerations make a pixel detector for such large scale tracking applications at present not feasible. Instead strip detectors are under development, which will provide excellent precision in a plane perpendicular to the electron-positron beam. Silicon microstrip detectors are extremely well understood devices, which have been used in large quantities in many experiments, most recently on an unprecedented scale by the LHC experiments. A typical detector fabricated with currently established technology might consist of a 300 μm thick layer of high resistivity silicon, with strips typically every 50 μm running the length of the detector. Charge is collected by the strips. These detectors measure one coordinate very well, with a precision of